#502514

order	512
nilpotency class	4
isoclinism	isoclinic class of #500857
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

682 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 409710);