#500856

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

949 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)(7,19)(8,24)(9,29)(10,16)(11,17)(12,15)(13,22)(14,26)(18,30)(20,23)(21,36)(25,34)(27,32)(28,35)(31,33);(5,6,12,29,11,18,31,34,10,16,25,33,30,17,9,15)(7,20,32,27,23,19,35,36,22,26,24,8,14,13,21,28);(2,4)(5,7)(6,13)(8,15)(9,21)(10,22)(11,23)(12,32)(14,30)(16,19)(17,26)(18,20)(24,25)(27,34)(28,29)(31,35)(33,36);(5,8,25,19,11,28,9,26,10,27,12,13,30,36,31,20)(6,14,33,35,18,7,15,24,16,23,29,21,17,22,34,32)

Polycyclic presentation:

2^9|b^2=eh,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, 408052);