#6769335

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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 ◁ #2293 ◁ 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)(19,30)(20,27)(21,35)(22,28)(23,29)(24,39)(25,33)(26,34)(31,41)(32,43)(36,45)(37,44)(38,42)(40,47)(46,48);(1,2)(3,6)(4,5)(7,8)(9,10)(11,16)(12,14)(13,15)(17,19)(18,25)(20,31)(21,36)(22,38)(23,33)(24,32)(26,40)(27,45)(28,43)(29,30)(34,48)(35,41)(37,47)(39,42)(44,46);(2,5)(6,8)(9,11,13,12)(10,14,15,16)(17,20,23,21)(18,26,29,44)(19,31,33,36)(22,34,24,37)(25,40,30,46)(27,28,35,39)(32,47,38,48)(41,42,45,43);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,14,15,16)(17,21,23,20)(18,27,29,35)(19,31,33,36)(22,34,24,37)(25,41,30,45)(26,28,44,39)(32,48,38,47)(40,42,46,43);(2,5)(6,8)(17,22,23,24)(18,28,29,39)(19,32,33,38)(20,34,21,37)(25,42,30,43)(26,35,44,27)(31,47,36,48)(40,45,46,41)

Polycyclic presentation:

2^9|a^2=fi,c^2=f,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=efh,g^c=gi,e^d=ei

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, 6676531);