#72307104

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#8350
commutator	C₂×M₁₆	C₂³×C₄
Frattini, cosocle	C₂×#131	C₂⁴
socle	C₂	C₂×#8350
prime radical	#10577575	C₂
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂×#131 ▷ C₂×C₄ ▷ C₂ ▷ C₁

4030 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:

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

Polycyclic presentation:

2^10|b^2=hj,d^2=fg,f^2=i,i^2=j,b^a=bj,c^b=cj,d^a=de,d^b=df,e^b=eh,f^a=fhj,f^b=fi,f^e=fj,g^a=ghj,g^e=gj,h^d=hj,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307130, #72307207, #72307633
maximal normal subgroup of	#72307130, #72307207, #72307633

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