#72225316

order	1024
nilpotency class	5
automorphism	?
homology	Abelianization (H₁) = C₈² · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#1333
commutator	C₁₆	C₈²
Frattini, cosocle	C₄×#346	C₂²
socle	C₂	#93270
prime radical	C₂³	#1389
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ C₄²×C₈ ◁ C₄×#346 ◁ G
LECS	G ▷ C₄×#346 ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

294 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 2 generators on 64 points:

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

Polycyclic presentation:

2^10|a^2=e,b^2=c,c^2=f,d^2=gij,e^2=h,g^2=ij,h^2=j,i^2=j,b^a=bd,c^a=ci,d^a=dg,d^b=dgij,d^c=dj,e^b=ej,f^a=fj,g^a=gi,g^b=gij,i^a=ij,i^b=ij

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225320, #72225331, #72225333, …
maximal normal subgroup of	#72225320, #72225331, #72225333, …

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