#495937

order	512
nilpotency class	4
isoclinism	isoclinic class of #33517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	#3050	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

590 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)(9,10,15,19,13,16,29,20)(11,21,25,37,23,40,26,35)(12,18,28,38,27,33,14,30)(17,34,36,22,32,31,39,24);(2,5)(6,8)(10,16)(17,32)(18,33)(19,20)(21,40)(30,38)(35,37)(36,39);(1,3,4,7)(2,6,5,8)(9,11,13,23)(10,17,16,32)(12,24,27,22)(14,31,28,34)(15,25,29,26)(18,37,33,35)(19,36,20,39)(21,38,40,30);(9,12,15,28,13,27,29,14)(10,18,19,38,16,33,20,30)(11,22,25,31,23,24,26,34)(17,35,36,21,32,37,39,40)

Polycyclic presentation:

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

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, 403133);