#500854

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#443 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=eh,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,g^c=gi

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, 408050);