#509175

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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

784 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416371);