#509164

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

797 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)(5,6,9,14,27,34,36,31,11,16,28,30,10,15,25,12)(7,13,18,32,17,24,35,26,20,8,21,22,19,23,33,29);(6,12)(7,17)(8,22)(9,25)(10,27)(13,32)(14,15)(16,31)(19,20)(23,26)(24,29)(28,36)(30,34)(33,35);(2,4)(5,7)(6,13)(8,16)(9,18)(10,19)(11,20)(12,29)(14,32)(15,23)(17,27)(21,28)(22,30)(24,34)(25,33)(26,31)(35,36);(2,4)(5,8,11,13)(6,7,16,20)(9,26,28,29)(10,23,27,24)(12,18,31,21)(14,33,30,35)(15,17,34,19)(22,36,32,25)

Polycyclic presentation:

2^9|a^2=e,d^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, 416360);