#72225319

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×#102
commutator	C₂×C₈	C₂×C₄×C₈
Frattini, cosocle	C₄²×C₈	C₂³
socle	C₂³	C₄×#102
prime radical	C₂²×D₈	C₄²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₄²×C₈ ◁ C₄×C₈² ◁ G
LECS	G ▷ C₄²×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

833 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 32 points:

(17,18)(19,28)(20,23)(21,32)(22,25)(24,31)(26,29)(27,30);(1,2,3,5,4,6,7,8)(10,13)(14,16)(18,23,25,24)(19,20)(21,30)(26,32,28,27)(29,31);(9,10,11,14,12,13,15,16)(17,19,21,20,22,29,30,31)(18,24,27,26,25,23,32,28)

Polycyclic presentation:

2^10|b^2=d,c^2=f,d^2=g,e^2=hj,f^2=i,h^2=j,b^a=be,c^a=chj,c^b=chij,d^a=dj,e^a=eh,e^b=ehj,f^a=fj,f^b=fj,h^a=hj,h^b=hj

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225335, #72227034, #72227065, …
maximal normal subgroup of	#72225335, #72227034, #72227065, …
maximal quotient of	#72225320, #72258160

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