#495456

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×C₄⋊C₈
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

572 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)(3,6)(4,5)(7,8)(9,10,14,11,13,18,27,22,15,19,29,23,28,17,33,21)(12,16,26,20,25,32,39,36,24,30,38,34,37,31,40,35);(9,11,14,18,13,22,27,19,15,23,29,17,28,21,33,10)(12,20,26,32,25,36,39,30,24,34,38,31,37,35,40,16);(2,5)(6,8)(9,12)(10,16)(11,20)(13,25)(14,26)(15,24)(17,31)(18,32)(19,30)(21,35)(22,36)(23,34)(27,39)(28,37)(29,38)(33,40);(1,3,4,7)(2,6,5,8)(12,24)(16,30)(20,34)(25,37)(26,38)(31,32)(35,36)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=f,e^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=ei,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402652);