#6907802

order	512
nilpotency class	3
isoclinism	isoclinic class of #6885129
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1313 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)(4,8)(5,7)(17,18,23,27,22,26,21,28)(19,24,36,38,30,40,31,39)(20,32,33,25,35,42,34,41)(29,43,45,37,44,48,46,47);(1,2,6,3)(4,7,5,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,31)(22,30)(23,36)(25,37)(26,40)(27,38)(28,39)(32,43)(33,45)(34,46)(35,44)(41,47)(42,48);(1,3,6,2)(4,7,5,8)(9,11)(10,13)(12,16)(14,15)(17,20)(18,25)(19,29)(21,33)(22,35)(23,34)(24,37)(26,41)(27,32)(28,42)(30,44)(31,45)(36,46)(38,43)(39,48)(40,47);(1,4,6,5)(2,7,3,8)(11,16)(13,15)(18,26)(19,30)(25,41)(27,28)(29,44)(31,36)(32,42)(45,46);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,27,26,28)(19,31,30,36)(20,33,35,34)(24,38,40,39)(25,32,41,42)(29,45,44,46)(37,43,47,48)

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=f,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dfi,e^a=ei,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6814998);