#509298

order	512
nilpotency class	4
isoclinism	isoclinic class of #509298
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#30567
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ #2930 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 416494);