#509155

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

828 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:

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

Polycyclic presentation:

2^9|a^2=e,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi

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, 416351);