#72307268

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#21541
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
socle	C₂	#376120
prime radical	C₂×#13309	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ #20385 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

2277 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:

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

Polycyclic presentation:

2^10|b^2=hj,c^2=ghij,d^2=e,e^2=j,g^2=i,i^2=j,b^a=bh,c^a=cf,c^b=cg,d^a=df,d^b=dfij,d^c=dfi,e^b=ej,e^c=ej,f^b=fj,g^b=gi,g^c=gj,h^c=hj,h^d=hj,i^b=ij

Conjugacy classes & character table

100 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307299, #72343936, #72343980
maximal normal subgroup of	#72307299, #72343936, #72343980

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