#501226

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×#3283	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

1102 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)(11,23)(12,18)(13,21)(14,29)(15,33)(16,22)(17,25)(19,28)(20,31)(24,34)(26,38)(27,36)(30,37)(32,39)(35,40);(5,6)(7,8)(9,11,16,12)(10,17,22,36)(13,26,15,24)(14,30,32,40)(18,29,23,39)(19,38,20,34)(21,37,33,35)(25,28,27,31);(2,4)(6,8)(9,12,16,11)(10,18,22,23)(13,26,15,24)(14,27,32,25)(17,29,36,39)(19,37,20,35)(21,38,33,34)(28,30,31,40);(9,13)(10,19)(11,24)(12,26)(15,16)(17,34)(18,37)(20,22)(21,29)(23,35)(28,31)(30,40)(33,39)(36,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^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, 408422);