#495385

order	512
nilpotency class	4
isoclinism	isoclinic class of #11677
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=i,e^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,h^a=hi,e^b=ei,g^b=gi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402581);