#6854447

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1246 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=hi,c^2=gh,d^2=fi,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^d=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, 6761643);