#509188

order	512
nilpotency class	4
isoclinism	isoclinic class of #509065
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#24266	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×D₄ ◁ C₂×C₈×D₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

805 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fh,h^a=hi,c^b=cg,d^b=de,f^c=fh,h^c=hi,f^d=fh,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416384);