#72280114

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#9001
commutator	#376	C₂²×C₄
Frattini, cosocle	C₂×#376	C₂³
socle	C₂	#99639
prime radical	#233619	C₂
LCS	G ▷ #376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂×#376 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^10|b^2=fh,c^2=e,d^2=ij,e^2=hi,f^2=hj,g^2=j,b^a=bf,c^a=cf,c^b=cd,d^a=dg,d^b=dgi,d^c=dgj,e^a=egh,e^b=egi,e^d=ej,f^a=fhj,f^c=fgj,f^d=fj,f^e=fj,g^a=gj,g^b=gj,h^c=hj,i^a=ij,i^b=ij,i^c=ij

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280134, #72280366
maximal normal subgroup of	#72280134, #72280366

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, 2, F.6*F.8);;
gap> SetPower(coll, 3, F.5);;
gap> SetPower(coll, 4, F.9*F.10);;
gap> SetPower(coll, 5, F.8*F.9);;
gap> SetPower(coll, 6, F.8*F.10);;
gap> SetPower(coll, 7, F.10);;
gap> SetConjugate(coll, 2, 1, F.2*F.6);;
gap> SetConjugate(coll, 3, 1, F.3*F.6);;
gap> SetConjugate(coll, 3, 2, F.3*F.4);;
gap> SetConjugate(coll, 4, 1, F.4*F.7);;
gap> SetConjugate(coll, 4, 2, F.4*F.7*F.9);;
gap> SetConjugate(coll, 4, 3, F.4*F.7*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.7*F.8);;
gap> SetConjugate(coll, 5, 2, F.5*F.7*F.9);;
gap> SetConjugate(coll, 5, 4, F.5*F.10);;
gap> SetConjugate(coll, 6, 1, F.6*F.8*F.10);;
gap> SetConjugate(coll, 6, 3, F.6*F.7*F.10);;
gap> SetConjugate(coll, 6, 4, F.6*F.10);;
gap> SetConjugate(coll, 6, 5, F.6*F.10);;
gap> SetConjugate(coll, 7, 1, F.7*F.10);;
gap> SetConjugate(coll, 7, 2, F.7*F.10);;
gap> SetConjugate(coll, 8, 3, F.8*F.10);;
gap> SetConjugate(coll, 9, 1, F.9*F.10);;
gap> SetConjugate(coll, 9, 2, F.9*F.10);;
gap> SetConjugate(coll, 9, 3, F.9*F.10);;
gap> UpdatePolycyclicCollector(coll);;
gap> G := GroupByRws(coll);