#501211

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
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×#3285	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

1105 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)(11,22)(12,18)(13,27)(14,29)(15,34)(16,21)(17,24)(19,28)(20,32)(23,38)(25,37)(26,39)(30,35)(31,36)(33,40);(5,6)(7,8)(9,11,16,13)(10,17,21,19)(12,23,15,25)(14,30,33,31)(18,35,34,36)(20,37,39,38)(22,40,27,29)(24,26,28,32);(2,4)(6,8)(9,12,16,15)(10,18,21,34)(11,23,13,25)(14,26,33,32)(17,35,19,36)(20,29,39,40)(22,38,27,37)(24,30,28,31);(9,13,16,11)(10,19,21,17)(12,25,15,23)(14,31,33,30)(18,36,34,35)(20,38,39,37)(22,29,27,40)(24,32,28,26)

Polycyclic presentation:

2^9|a^2=f,b^2=i,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,e^b=eh,h^b=hi,e^d=eh,h^d=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408407);