#502198

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
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

741 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:

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

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,d^b=di,e^b=eh,h^b=hi,f^c=fi,g^c=gi,e^d=eh,h^d=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409394);