#502185

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
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

777 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)(7,14)(8,23)(9,13)(10,17)(11,28)(12,18)(15,22)(16,25)(19,34)(20,31)(21,30)(24,33)(26,35)(27,32)(29,36);(1,3,4,2)(6,13)(7,19)(9,25)(10,11)(14,33)(15,23)(16,17)(18,32)(20,27)(21,29)(22,26)(28,31)(30,34)(35,36);(2,3)(5,7)(6,14)(8,20)(9,21)(10,26)(11,19)(12,22)(13,30)(15,18)(16,33)(17,35)(23,31)(24,25)(27,29)(28,34)(32,36);(5,8)(6,15)(7,20)(9,26)(10,21)(11,29)(12,24)(13,23)(14,18)(16,36)(17,35)(19,27)(22,25)(28,34)(30,31)(32,33)

Polycyclic presentation:

2^9|a^2=f,b^2=f,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,e^b=eh,g^b=gi,h^b=hi,g^c=gi,e^d=eh,h^d=hi

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, 409381);