#509163

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

802 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)(5,6,9,14,28,31,19,36,10,15,29,34,11,16,30,27)(7,17,20,33,22,13,8,24,21,12,26,32,23,18,25,35);(5,6)(7,18)(8,24)(9,27)(10,15)(11,31)(12,22)(13,21)(14,30)(16,28)(17,23)(19,34)(20,32)(25,35)(26,33)(29,36);(2,4)(5,7)(6,12)(8,19)(9,20)(10,21)(11,23)(13,16)(14,32)(15,17)(18,31)(22,28)(24,27)(25,30)(26,29)(33,34)(35,36);(2,4)(5,8)(6,13)(7,19)(9,22)(10,25)(11,26)(12,16)(14,33)(15,18)(17,31)(20,28)(21,30)(23,29)(24,27)(32,34)(35,36)

Polycyclic presentation:

2^9|a^2=e,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, 416359);