#6793181

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
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

1402 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:

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

Polycyclic presentation:

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

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, 6700377);