#495389

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×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

868 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)(9,10,13,19,16,22,15,21)(11,23,28,40,29,39,26,35)(12,30,24,18,31,20,14,32)(17,34,37,27,38,25,36,33);(2,5)(6,7)(9,11,24,27,15,28,12,25,16,29,14,33,13,26,31,34)(10,17,20,39,19,36,32,40,22,38,30,23,21,37,18,35);(1,3,4,8)(2,6,5,7)(9,12,13,24,16,31,15,14)(10,18,21,30,22,32,19,20)(11,25,26,27,29,34,28,33)(17,35,37,23,38,40,36,39);(9,13,16,15)(10,19,22,21)(11,26,29,28)(12,24,31,14)(17,36,38,37)(18,20,32,30)(23,35,39,40)(25,27,34,33)

Polycyclic presentation:

2^9|a^2=g,b^2=efg,c^2=f,d^2=i,e^2=h,f^2=i,g^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, 402585);