#501223

order	512
nilpotency class	4
isoclinism	isoclinic class of #499724
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3287	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

1128 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)(11,17)(12,23)(13,16)(14,20)(15,29)(18,25)(19,26)(21,38)(22,31)(24,34)(27,33)(28,32)(30,37)(35,39)(36,40);(5,6)(7,8)(10,16)(11,15)(12,24)(13,26)(17,19)(18,34)(20,32)(22,28)(23,35)(25,40)(27,39)(29,31)(33,38)(36,37);(2,4)(6,8)(9,11,14,15)(10,17,20,29)(12,21,24,30)(13,22,28,26)(16,31,32,19)(18,33,35,36)(23,38,34,37)(25,27,39,40);(9,12,14,24)(10,18,20,35)(11,21,15,30)(13,27,28,40)(16,23,32,34)(17,33,29,36)(19,37,31,38)(22,39,26,25)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408419);