#72320926

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×#102
commutator	C₂×C₈	C₂³×C₈
Frattini, cosocle	C₈²	C₂⁴
socle	C₂²	C₂²×D₄⋊C₈
prime radical	C₂×#576	C₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₈×#41 ◁ C₈×#131 ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

1419 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)(19,24)(20,23)(21,25)(22,27)(26,34)(28,33)(29,38)(30,35)(31,39)(32,36)(37,45)(40,44)(41,43)(42,46)(47,48);(1,2,3,7,4,8,9,14,5,6,10,12,11,13,15,16)(18,23,25,26,27,36,28,37)(20,32)(24,35,39,40,38,46,41,47)(30,42)(34,45)(44,48);(2,6)(7,12)(8,13)(14,16)(17,19)(18,24)(20,30)(21,31)(22,29)(23,35)(25,39)(26,40)(27,38)(28,41)(32,42)(33,43)(34,44)(36,46)(37,47)(45,48);(19,29)(24,38)(30,42)(31,43)(35,46)(39,41)(40,47)(44,48)

Polycyclic presentation:

2^10|b^2=e,e^2=g,f^2=hj,g^2=i,h^2=j,b^a=bf,c^b=ci,d^c=dj,e^a=eh,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, #72321060, #72321655
maximal normal subgroup of	#72321010, #72321060, #72321655

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