#72321332

order	1024
nilpotency class	3
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₄²	C₂⁴
socle	C₂⁴	C₂×C₄×D₄
prime radical	C₂×#1999	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

4277 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 24 points:

(5,6)(7,8)(17,18,19,21,20,23,24,22);(2,4)(9,10,12,15,13,16,11,14)(18,21)(19,24)(22,23);(1,2,3,4)(6,8)(9,11,13,12)(10,14,16,15)(18,22)(19,24)(21,23);(2,4)(18,22)(19,24)(21,23)

Polycyclic presentation:

2^10|a^2=h,b^2=e,c^2=fi,e^2=i,g^2=j,h^2=j,b^a=bhj,c^a=cg,c^b=cf,d^a=dh,d^c=df,g^b=gj,g^c=gj,g^d=gj,h^b=hj,h^c=hj,h^d=hj

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321430, #72321510, #72321757, …
maximal normal subgroup of	#72321430, #72321510, #72321757, …
maximal quotient of	#72322629

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