#6802611

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	#25723	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1781 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)(17,18,22,27,21,26,35,24)(19,28,31,37,30,23,36,39)(20,32,34,40,33,25,41,42)(29,38,45,48,44,43,46,47);(2,5)(3,7)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,29)(21,30)(22,31)(24,37)(25,38)(26,28)(27,39)(32,43)(33,44)(34,45)(35,36)(40,47)(41,46)(42,48);(11,16)(13,15)(18,24)(22,35)(23,37)(25,40)(26,27)(28,39)(31,36)(32,42)(34,41)(38,47)(43,48)(45,46);(2,5)(6,8)(10,14)(13,15)(17,20,21,33)(18,25,26,32)(19,29,30,44)(22,34,35,41)(23,38,28,43)(24,40,27,42)(31,45,36,46)(37,47,39,48);(1,3,4,7)(2,6,5,8)(19,30)(20,33)(23,28)(25,32)(31,36)(34,41)(37,39)(40,42)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6709807);