#6579586

order	512
nilpotency class	3
isoclinism	isoclinic class of D₈×#120
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×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1511 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)(11,13)(12,14)(15,16)(17,18,21,22)(19,23,27,25)(20,24,29,30)(26,31,32,28);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,19,21,27)(18,23,22,25)(20,28,29,31)(24,26,30,32);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,22)(19,25)(23,27)(24,30)(26,28)(31,32);(17,19,21,27)(18,23,22,25)(20,28,29,31)(24,26,30,32);(17,20,21,29)(18,24,22,30)(19,26,27,32)(23,31,25,28)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=f,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=eh,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486782);