#72308247

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#21324
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#362708
prime radical	C₂²×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ C₂×#2875 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

3855 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=f,b^2=f,c^2=f,d^2=g,e^2=i,g^2=i,h^2=j,i^2=j,b^a=bfj,c^b=cfj,d^a=dh,d^b=de,d^c=dij,e^a=ej,e^b=ei,f^d=fj,g^a=gj,g^b=gi,g^c=gj,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308342
maximal normal subgroup of	#72308342

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