#502197

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

758 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,d^b=di,e^b=eh,h^b=hi,f^c=fi,g^c=gi,e^d=eh,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, 409393);