#6818192

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1431 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)(4,5)(7,8)(9,10,12,14)(11,15,16,13)(20,31)(24,39)(29,43)(32,46)(34,40)(38,44)(42,48)(45,47);(2,5)(3,7)(17,18,22,26)(19,27,28,41)(20,32,31,46)(21,35,33,25)(23,36,37,30)(24,40,39,34)(29,44,43,38)(42,47,48,45);(9,11,12,16)(10,13,14,15)(17,19,21,30,22,28,33,36)(18,23,25,27,26,37,35,41)(20,29,34,45,31,43,40,47)(24,38,32,42,39,44,46,48);(2,5)(6,8)(19,28)(23,37)(27,41)(29,43)(30,36)(38,44)(42,48)(45,47);(1,3,4,7)(2,6,5,8)(17,20,33,40,22,31,21,34)(18,24,35,46,26,39,25,32)(19,29,36,47,28,43,30,45)(23,38,41,48,37,44,27,42)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6725388);