#509169

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
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

801 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)(5,6,13,20,32,21,22,33,11,7,10,16,12,17,34,31)(8,18,30,35,25,19,9,27,24,14,23,36,26,15,29,28);(5,7,11,6)(8,19,24,15)(9,28,29,27)(10,31,13,33)(12,17,32,21)(14,25,18,26)(16,22,20,34)(23,36,30,35);(2,4)(5,8)(6,14)(7,18)(9,22)(10,23)(11,24)(12,26)(13,30)(15,21)(16,35)(17,19)(20,36)(25,32)(27,31)(28,33)(29,34);(2,4)(5,9)(6,15)(7,19)(8,22)(10,26)(11,29)(12,23)(13,25)(14,21)(16,35)(17,18)(20,36)(24,34)(27,33)(28,31)(30,32)

Polycyclic presentation:

2^9|a^2=ei,b^2=i,e^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=defi,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416365);