#502194

order	512
nilpotency class	4
isoclinism	isoclinic class of #500730
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

743 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6,10,16,12,18,11,17)(7,19,23,34,25,14,24,33)(8,26,22,13,28,20,27,32)(9,29,30,36,31,35,21,15);(1,2)(3,4)(5,7,10,23,12,25,11,24)(6,13,16,20,18,32,17,26)(8,21,22,9,28,30,27,31)(14,15,33,29,19,36,34,35);(2,4)(5,6)(7,20)(8,14)(9,15)(10,17)(11,16)(12,18)(13,23)(19,28)(21,29)(22,34)(24,32)(25,26)(27,33)(30,35)(31,36);(5,8,12,28)(6,14,18,19)(7,21,25,30)(9,24,31,23)(10,22,11,27)(13,15,32,36)(16,33,17,34)(20,29,26,35)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,h^a=hi,c^b=cg,e^b=eh,h^b=hi,f^c=fi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409390);