#360947

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

794 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268143);