#72341007

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#46996
commutator, Frattini, cosocle	C₂⁵	C₂⁵
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2360 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 64 points:

(1,2)(3,8,15,26)(4,9,13,24)(5,18)(6,20)(7,12)(10,16)(11,23)(14,31,21,25)(17,27,22,29)(19,30)(28,32)(33,34)(35,40,47,57)(36,48)(37,53)(38,43,54,60)(39,44)(41,52)(42,46)(45,56,49,63)(50,58)(51,59,62,61)(55,64);(1,3,7,15)(2,8)(4,16,13,5)(6,21)(9,18)(10,24)(11,25,20,31)(12,26)(14,23)(17,32)(19,22)(27,28,29,30)(33,35)(34,40,44,57)(36,49)(37,54,46,38)(39,47)(41,56,48,63)(42,60)(43,53)(45,52)(50,51,55,62)(58,61)(59,64);(1,4)(2,9,12,24)(3,5)(6,22,23,17)(7,13)(8,18,26,10)(11,29)(14,32,21,19)(15,16)(20,27)(25,28)(30,31)(33,36)(34,41,44,48)(35,45)(37,55,46,50)(38,51,54,62)(39,52)(40,56,57,63)(42,58)(43,61)(47,49)(53,64)(59,60);(1,5)(2,10)(3,13,15,4)(6,19)(7,16)(8,24,26,9)(11,30)(12,18)(14,17,21,22)(20,28)(23,32)(25,29,31,27)(33,37,39,46)(34,42,44,53)(35,38)(36,50,52,55)(40,43)(41,58,48,64)(45,62)(47,54)(49,51)(56,61)(57,60)(59,63);(1,6,7,23)(2,11,12,20)(3,14)(4,17,13,22)(5,19)(8,25)(9,27,24,29)(10,28)(15,21)(16,32)(18,30)(26,31)(33,38)(34,43,44,60)(35,46)(36,51,52,62)(37,47)(39,54)(40,53,57,42)(41,59)(45,55,49,50)(48,61)(56,64)(58,63)

Polycyclic presentation:

2^10|a^2=f,b^2=gij,c^2=g,d^2=fj,e^2=fghij,b^a=bfg,c^a=ch,c^b=cgi,d^a=dfi,d^b=dhj,d^c=dfghij,e^a=eghij,e^b=efgij,e^c=egh,e^d=efgij,f^b=fi,f^c=fij,f^e=fj,g^a=gi,g^d=gj,g^e=gi,h^a=hij,h^b=hj,h^c=hi,h^d=hi,h^e=hij

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72341133, #72341188
maximal normal subgroup of	#72341133, #72341188

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from its polycyclic presentation:

gap> F := FreeGroup(IsSyllableWordsFamily, "a", "b", "c", "d", "e", "f", "g", "h", "i", "j");;
gap> coll := SingleCollector(F, [2, 2, 2, 2, 2, 2, 2, 2, 2, 2]);;
gap> SetPower(coll, 1, F.6);;
gap> SetPower(coll, 2, F.7*F.9*F.10);;
gap> SetPower(coll, 3, F.7);;
gap> SetPower(coll, 4, F.6*F.10);;
gap> SetPower(coll, 5, F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6*F.7);;
gap> SetConjugate(coll, 3, 1, F.3*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.7*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 2, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.9*F.10);;
gap> SetConjugate(coll, 6, 2, F.6*F.9);;
gap> SetConjugate(coll, 6, 3, F.6*F.9*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.9);;
gap> SetConjugate(coll, 7, 4, F.7*F.10);;
gap> SetConjugate(coll, 7, 5, F.7*F.9);;
gap> SetConjugate(coll, 8, 1, F.8*F.9*F.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.9);;
gap> SetConjugate(coll, 8, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);