#72341041

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

2310 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)(3,11)(4,8)(5,18,19,9)(6,10)(7,14)(12,22)(13,21,27,25)(15,20)(16,23,29,30)(17,24)(26,32,28,31)(33,34)(35,39)(36,40,47,54)(37,41)(38,42)(43,58,60,51)(44,61)(45,53)(46,56,49,55)(48,52)(50,57)(59,64,62,63);(1,3)(2,7)(4,15)(5,13,19,27)(6,14)(8,22)(9,25,18,21)(10,11)(12,17)(16,28,29,26)(20,24)(23,31,30,32)(33,35)(34,39)(36,43,47,60)(37,48)(38,45)(40,51,54,58)(41,52)(42,53)(44,50)(46,59,49,62)(55,64,56,63)(57,61);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,23)(10,24)(11,22)(13,26)(14,15)(18,30)(19,29)(21,31)(25,32)(27,28)(40,54)(46,49)(51,58)(59,62);(1,3)(2,7,10,11)(4,15,17,12)(5,13)(6,14)(8,20)(9,25,18,21)(16,26,29,28)(19,27)(22,24)(23,31)(30,32)(33,36,38,47)(34,40,42,54)(35,43)(37,49,50,46)(39,51)(41,56,57,55)(44,59)(45,60)(48,62)(52,63)(53,58)(61,64);(1,5)(2,9,10,18)(3,13)(4,16,17,29)(6,19)(7,21,11,25)(8,23)(12,26,15,28)(14,27)(20,31)(22,32)(24,30)(33,37,38,50)(34,41,42,57)(35,44)(36,46,47,49)(39,52)(40,55,54,56)(43,59)(45,48)(51,63)(53,61)(58,64)(60,62)

Polycyclic presentation:

2^10|a^2=g,b^2=g,d^2=fgj,e^2=fgj,b^a=bgi,c^a=cg,c^b=ci,d^a=dh,d^b=df,d^c=dfghj,e^a=efi,e^b=efhij,e^c=egi,e^d=efgj,f^a=fi,f^b=fj,f^c=fi,f^d=fj,f^e=fi,g^d=gj,g^e=gi,h^a=hj,h^b=hij,h^c=hi,h^d=hi,h^e=hij

Conjugacy classes & character table

58 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.7);;
gap> SetPower(coll, 2, F.7);;
gap> SetPower(coll, 4, F.6*F.7*F.10);;
gap> SetPower(coll, 5, F.6*F.7*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7);;
gap> SetConjugate(coll, 3, 2, F.3*F.9);;
gap> SetConjugate(coll, 4, 1, F.4*F.8);;
gap> SetConjugate(coll, 4, 2, F.4*F.6);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.8*F.9*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.9);;
gap> SetConjugate(coll, 6, 2, F.6*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.9);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*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.10);;
gap> SetConjugate(coll, 8, 2, F.8*F.9*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);