#6819088

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806647
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1364 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=f,e^2=hi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6726284);