#6769435

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
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

1291 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fi,d^2=hi,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,g^c=gi

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, 6676631);