#6909304

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907909
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	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=gi,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,d^c=dfh

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, 6816500);