#509160

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

791 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)(5,6,12,28,27,35,25,13,11,17,9,15,10,16,29,32)(7,14,30,22,18,24,34,31,21,8,19,33,20,23,36,26);(6,13)(7,18)(8,22)(9,25)(10,27)(12,29)(14,33)(15,16)(17,32)(20,21)(23,26)(24,31)(28,35)(34,36);(2,4)(5,7)(6,14)(8,17)(9,19)(10,20)(11,21)(12,30)(13,31)(15,33)(16,23)(18,27)(22,28)(24,35)(25,34)(26,32)(29,36);(2,4)(5,8,11,14)(6,7,17,21)(9,26,12,31)(10,23,27,24)(13,19,32,30)(15,34,28,36)(16,18,35,20)(22,29,33,25)

Polycyclic presentation:

2^9|a^2=ei,d^2=i,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, 416356);