#502023

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33611	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

517 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)(3,5)(4,6)(7,8)(9,10)(11,22)(12,25)(13,31)(14,19)(15,21)(16,17)(18,23)(20,32)(24,37)(26,30)(27,29)(28,35)(33,38)(34,36)(39,40);(1,3,4,7)(2,5,6,8)(9,11,23,37,14,26,33,28,15,27,40,36,32,25,13,17)(10,16,31,12,20,34,39,29,21,35,38,30,19,24,18,22);(2,6)(5,8)(9,10)(11,24)(12,28)(13,18)(14,20)(15,21)(16,26)(17,30)(19,32)(22,37)(23,31)(25,35)(27,34)(29,36)(33,39)(38,40);(9,12,23,34,14,29,33,35,15,30,40,24,32,22,13,16)(10,17,31,11,20,37,39,26,21,28,38,27,19,36,18,25)

Polycyclic presentation:

2^9|b^2=efh,d^2=eh,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409219);