#6761326

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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

1272 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)(17,18,23,27,22,26,21,35)(19,28,31,39,30,24,36,40)(20,32,33,41,34,42,37,25)(29,44,47,38,45,48,46,43);(1,3,6,2)(4,8,5,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,29)(21,36)(22,30)(23,31)(25,38)(26,28)(27,40)(32,48)(33,47)(34,45)(35,39)(37,46)(41,43)(42,44);(9,11,12,16)(10,13,14,15)(17,20)(18,25)(19,29)(21,33)(22,34)(23,37)(24,38)(26,41)(27,42)(28,43)(30,45)(31,46)(32,35)(36,47)(39,48)(40,44);(1,4,6,5)(2,7,3,8)(18,26)(19,30)(25,41)(27,35)(29,45)(31,36)(32,42)(46,47);(1,5,6,4)(2,7,3,8)(10,14)(13,15)(17,21,22,23)(18,27,26,35)(19,31,30,36)(20,33,34,37)(24,39,28,40)(25,42,41,32)(29,46,45,47)(38,48,43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6668522);