#503276

order	512
nilpotency class	4
isoclinism	isoclinic class of #500854
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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 ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1081 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=ehi,c^2=f,d^2=ehi,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, 410472);