#72341028

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

2308 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)(3,10)(4,9)(5,8)(6,19,20,11)(7,12)(13,21)(14,22,18,26)(15,25)(16,23,30,27)(17,24)(28,31,29,32)(33,34)(35,44)(36,41,38,53)(37,50,52,42)(39,43)(40,48)(45,57,47,55)(46,56,51,60)(49,59)(54,58)(61,63)(62,64);(1,3)(2,8)(4,15)(5,7)(6,18,20,14)(9,21)(10,12)(11,22,19,26)(13,17)(16,29,30,28)(23,32,27,31)(24,25)(33,35)(34,40)(36,47,38,45)(37,51,52,46)(39,48)(41,55,53,57)(42,56,50,60)(43,44)(49,61)(54,62)(58,63)(59,64);(1,4,7,17)(2,9,12,24)(3,13)(5,15)(6,16)(8,21)(10,25)(11,27)(14,29,18,28)(19,23)(20,30)(22,31,26,32)(33,36)(34,41,43,53)(35,45)(37,49,52,54)(38,39)(40,55,44,57)(42,58)(46,62,51,61)(47,48)(50,59)(56,64)(60,63);(1,5,7,3)(2,10)(4,15)(6,18)(8,12)(9,21,24,25)(11,22,19,26)(13,17)(14,20)(16,28,30,29)(23,31)(27,32)(33,37)(34,42,43,50)(35,46,48,51)(36,49,38,54)(39,52)(40,56)(41,58)(44,60)(45,61)(47,62)(53,59)(55,63,57,64);(1,6)(2,11,12,19)(3,14)(4,16,17,30)(5,18)(7,20)(8,22,10,26)(9,23)(13,28,15,29)(21,31)(24,27)(25,32)(33,38,39,36)(34,41)(35,47)(37,49)(40,57,44,55)(42,59,50,58)(43,53)(45,48)(46,62,51,61)(52,54)(56,64)(60,63)

Polycyclic presentation:

2^10|a^2=g,b^2=g,c^2=hij,d^2=fgij,e^2=fj,b^a=bgi,c^a=cgh,c^b=cf,d^a=dhj,d^b=dfi,d^c=dfgh,e^a=efgi,e^b=efghj,e^c=efhj,e^d=efghi,f^a=fi,f^b=fj,f^c=fij,f^d=fj,g^c=gj,g^d=gj,g^e=gi,h^a=hj,h^b=hij,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, 3, F.8*F.9*F.10);;
gap> SetPower(coll, 4, F.6*F.7*F.9*F.10);;
gap> SetPower(coll, 5, F.6*F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.7*F.9);;
gap> SetConjugate(coll, 3, 1, F.3*F.7*F.8);;
gap> SetConjugate(coll, 3, 2, F.3*F.6);;
gap> SetConjugate(coll, 4, 1, F.4*F.8*F.10);;
gap> SetConjugate(coll, 4, 2, F.4*F.6*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.6*F.7*F.8);;
gap> SetConjugate(coll, 5, 1, F.5*F.6*F.7*F.9);;
gap> SetConjugate(coll, 5, 2, F.5*F.6*F.7*F.8*F.10);;
gap> SetConjugate(coll, 5, 3, F.5*F.6*F.8*F.10);;
gap> SetConjugate(coll, 5, 4, F.5*F.6*F.7*F.8*F.9);;
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*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 7, 3, F.7*F.10);;
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, 4, F.8*F.9);;
gap> SetConjugate(coll, 8, 5, F.8*F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);