#6854020

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1297 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)(17,18,22,24)(19,28,29,43)(20,33,35,26)(21,27,23,37)(25,31,38,32)(30,45,44,42)(34,41,36,40)(39,46,48,47);(2,5)(3,7)(9,11,12,16)(10,13,14,15)(18,24)(19,29)(26,33)(27,37)(30,44)(31,32)(40,41)(46,47);(10,14)(11,16)(17,19,23,32,22,29,21,31)(18,25,27,28,24,38,37,43)(20,30,34,46,35,44,36,47)(26,39,41,42,33,48,40,45);(2,5)(6,8)(10,14)(13,15)(17,20)(18,26)(19,30)(21,36)(22,35)(23,34)(24,33)(25,39)(27,41)(28,42)(29,44)(31,47)(32,46)(37,40)(38,48)(43,45);(1,3,4,7)(2,6,5,8)(17,21,22,23)(18,27,24,37)(19,31,29,32)(20,34,35,36)(25,28,38,43)(26,40,33,41)(30,46,44,47)(39,45,48,42)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,e^2=fhi,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, 6761216);