#362562

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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

766 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)(3,6)(4,5)(7,8)(9,10,13,16,15,20,14,19)(11,21,22,17,24,30,29,32)(12,25,27,18,26,31,28,34)(23,37,36,35,39,33,38,40);(2,5)(6,8)(10,16,20,19)(11,22,24,29)(12,26)(17,32)(18,25,34,31)(21,30)(23,38,39,36)(27,28);(1,3,4,7)(2,6,5,8)(9,11,13,22,15,24,14,29)(10,17,19,21,20,32,16,30)(12,23,27,36,26,39,28,38)(18,33,25,35,34,37,31,40);(2,5)(6,8)(9,12,13,27,15,26,14,28)(10,18,19,25,20,34,16,31)(11,23,22,36,24,39,29,38)(17,33,21,35,32,37,30,40)

Polycyclic presentation:

2^9|a^2=h,b^2=fg,c^2=e,d^2=f,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,d^b=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, 269758);