#360946

order	512
nilpotency class	3
isoclinism	isoclinic class of #24289
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

804 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)(9,10,15,20,16,21,14,34)(11,22,26,36,27,35,25,17)(12,28,24,38,31,37,30,18)(13,32,23,39,29,40,33,19);(2,5)(6,7)(9,11,14,25,16,27,15,26)(10,17,20,22,21,36,34,35)(12,29,30,23,31,13,24,33)(18,32,28,39,38,40,37,19);(9,12,14,30,16,31,15,24)(10,18,20,28,21,38,34,37)(11,23,25,13,27,33,26,29)(17,19,22,32,36,39,35,40);(1,3,4,8)(2,6,5,7)(9,13,14,33,16,29,15,23)(10,19,20,32,21,39,34,40)(11,24,25,12,27,30,26,31)(17,18,22,28,36,38,35,37)

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,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 268142);