#72320562

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#442
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₈²	C₂⁴
socle	C₂²	C₂×#1580
prime radical	C₂×#508	C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #11402 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

612 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

(17,18)(19,32)(20,27)(21,40)(22,42)(23,29)(24,43)(25,31)(26,35)(28,37)(30,41)(33,47)(34,48)(36,44)(38,45)(39,46);(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16)(17,19,22,34,24,36,23,38,25,37,41,21,20,33,39,35)(18,26,29,44,27,40,42,32,31,45,46,47,43,48,30,28);(2,6)(7,12)(8,13)(14,16)(17,20,25,24)(18,27,31,43)(19,33,37,36)(21,38,34,35)(22,39,41,23)(26,40,45,48)(28,44,32,47)(29,42,46,30);(17,21,41,33,24,35,39,19,25,34,22,36,20,38,23,37)(18,28,29,45,27,44,42,48,31,32,46,26,43,47,30,40)

Polycyclic presentation:

2^10|b^2=e,c^2=j,d^2=fh,e^2=g,f^2=hj,g^2=i,h^2=j,b^a=bf,c^b=cij,d^a=df,d^b=dh,e^a=eh,e^d=ej,f^a=fh,f^b=fj,g^a=gj,h^a=hj

Conjugacy classes & character table

124 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320600, #72321060, #72321542, …
maximal normal subgroup of	#72320600, #72321060, #72321542, …

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