#6450860

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437537
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1721 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=f,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=dh,e^b=ei,h^b=hi,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6358056);