#6867953

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	#31066	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1519 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=i,d^2=h,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,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, 6775149);