#509167

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

Polycyclic presentation:

2^9|a^2=ei,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, 416363);