#6851468

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1254 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)(17,18,22,28)(19,29,32,24)(20,33,35,26)(21,27,23,38)(25,37,43,34)(30,40,47,45)(31,42,36,41)(39,48,44,46);(1,2)(3,6)(4,5)(7,8)(9,10,13,16)(11,15,12,14)(17,19,22,32)(18,24,28,29)(20,30,35,47)(21,36,23,31)(25,39,43,44)(26,40,33,45)(27,42,38,41)(34,48,37,46);(9,11,13,12)(10,14,16,15)(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,39)(26,27)(28,43)(29,44)(31,46)(32,47)(33,38)(36,48)(40,42)(41,45);(1,3,4,7)(2,6,5,8)(17,20)(18,26)(19,30)(21,37)(22,35)(23,34)(24,40)(25,27)(28,33)(29,45)(31,48)(32,47)(36,46)(38,43)(39,42)(41,44);(2,5)(6,8)(9,12,13,11)(10,15,16,14)(17,21,22,23)(18,27,28,38)(19,31,32,36)(20,34,35,37)(24,41,29,42)(25,26,43,33)(30,46,47,48)(39,45,44,40)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=h,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,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, 6758664);