#72320967

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×#509	C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ #11410 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

655 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^10|b^2=e,c^2=f,e^2=g,f^2=hj,g^2=i,h^2=j,b^a=bf,c^a=cfh,c^b=chij,d^a=dj,d^c=dj,e^a=eh,e^c=ej,f^a=fh,f^b=fj,g^a=gj,h^a=hj

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321010, #72321171, #72321716
maximal normal subgroup of	#72321010, #72321171, #72321716

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.6);;
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, 1, F.3*F.6*F.8);;
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, 4, 3, F.4*F.10);;
gap> SetConjugate(coll, 5, 1, F.5*F.8);;
gap> SetConjugate(coll, 5, 3, 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);