#72317628

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	#9497
commutator	C₂⁴×C₄	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×#41	C₂³
socle	C₂	#120014
LCS	G ▷ C₂⁴×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂³×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1694 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:

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

Polycyclic presentation:

2^10|a^2=e,b^2=ei,c^2=d,d^2=j,e^2=j,b^a=bg,c^a=cf,c^b=cej,d^a=dh,d^b=dj,f^b=fi,f^c=fh,f^d=fj,g^c=gi,g^d=gj,h^b=hj,h^c=hj,i^c=ij

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72317635, #72317737, #72317779
maximal normal subgroup of	#72317635, #72317737, #72317779

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