#500859

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
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

947 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)(7,14)(8,23)(9,30)(10,17)(11,32)(12,18)(13,16)(15,26)(19,21)(20,27)(22,34)(24,25)(28,35)(29,33)(31,36);(5,6,13,30,11,17,31,24,12,18,25,36,10,32,9,16)(7,19,33,29,21,14,35,27,22,26,23,8,15,34,20,28);(2,4)(5,7)(6,14)(8,24)(9,20)(10,15)(11,21)(12,22)(13,33)(16,29)(17,26)(18,34)(19,32)(23,25)(27,30)(28,36)(31,35);(5,8,25,14,11,28,9,26,12,29,13,34,10,27,31,19)(6,15,36,35,17,7,16,23,18,21,30,20,32,22,24,33)

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,d^c=di,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408055);