#72307235

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#21233
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#361594
prime radical	#10585859	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20384 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

5127 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:

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

Polycyclic presentation:

2^10|a^2=j,b^2=e,c^2=ij,d^2=egj,f^2=j,g^2=i,i^2=j,b^a=bh,c^a=cij,c^b=cf,d^a=dfgij,d^b=dgi,d^c=dfi,e^c=ej,g^a=gi,g^b=gij,g^c=gj,h^c=hj,h^d=hj,i^a=ij,i^b=ij

Conjugacy classes & character table

94 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307299, #72320238, #72320639
maximal normal subgroup of	#72307299, #72320238, #72320639

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