#294312

order	512
nilpotency class	3
isoclinism	isoclinic class of #267659
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1002 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,16)(13,14)(17,18,22,25)(19,23,21,30)(20,27,29,32)(24,26,31,28);(2,5)(6,7)(11,16)(12,15)(17,19,22,21)(18,23,25,30)(20,26,29,28)(24,27,31,32);(9,11,15,10,13,16,12,14)(17,20,19,26,22,29,21,28)(18,24,30,32,25,31,23,27);(1,3,4,8)(2,6,5,7)

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=fh,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^c=gi

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, 201508);