#6764598

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1470 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)(11,13)(12,14)(15,16)(17,18,22,26)(19,27,29,43)(20,24,31,39)(21,34,35,25)(23,36,38,30)(28,42,32,46)(33,48,44,41)(37,47,40,45);(2,5)(3,7)(9,11)(10,13)(12,15)(14,16)(20,31)(24,39)(28,32)(33,44)(37,40)(41,48)(42,46)(45,47);(10,14)(11,15)(17,19,21,30,22,29,35,36)(18,23,25,27,26,38,34,43)(20,32,33,47,31,28,44,45)(24,40,41,46,39,37,48,42);(2,5)(6,8)(17,20,22,31)(18,24,26,39)(19,28,29,32)(21,33,35,44)(23,37,38,40)(25,41,34,48)(27,42,43,46)(30,45,36,47);(1,3,4,7)(2,6,5,8)(11,15)(13,16)(19,29)(20,31)(23,38)(24,39)(27,43)(30,36)(33,44)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6671794);