#509173

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

796 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:

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

Polycyclic presentation:

2^9|a^2=e,c^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, 416369);