#6802666

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#25727	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1776 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6709862);