#6909136

order	512
nilpotency class	3
isoclinism	isoclinic class of #6907993
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1469 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)(19,28)(20,25)(21,26)(22,37)(23,27)(24,31)(29,40)(30,45)(32,48)(33,42)(34,44)(35,38)(36,43)(39,47)(41,46);(1,3,4,8)(2,5,6,7)(9,10)(11,15)(12,14)(13,16)(19,29)(24,38)(28,40)(30,32)(31,35)(39,41)(45,48)(46,47);(3,8)(5,7)(9,11)(10,13)(12,16)(14,15)(17,19,21,35,23,29,22,31)(18,24,37,40,27,38,26,28)(20,32,36,47,34,30,33,46)(25,41,42,45,44,39,43,48);(2,6)(5,7)(11,16)(13,15)(17,20,23,34)(18,25,27,44)(19,30,29,32)(21,36,22,33)(24,39,38,41)(26,43,37,42)(28,45,40,48)(31,47,35,46);(10,14)(13,15)(17,21,23,22)(18,26,27,37)(19,31,29,35)(20,33,34,36)(24,40,38,28)(25,42,44,43)(30,46,32,47)(39,45,41,48)

Polycyclic presentation:

2^9|a^2=f,b^2=f,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,e^b=efhi,d^c=dfh,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6816332);