#509174

order	512
nilpotency class	4
isoclinism	isoclinic class of #509097
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

779 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=i,d^2=i,e^2=hi,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,e^b=eh,h^b=hi,f^c=fi,e^d=eh,f^d=fi,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, 416370);