#6579663

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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 ◁ C₄×#126 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1461 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)(18,19)(23,25)(24,26)(27,28);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,13,14,16)(17,18)(19,21)(20,27)(22,23)(24,30)(25,31)(26,32)(28,29);(1,4,5,3)(2,6,7,8)(10,14)(13,16)(18,23)(19,25)(22,31)(24,27)(26,28)(30,32);(17,19,21,18)(20,28,29,27)(22,25,31,23)(24,30,26,32);(17,20,21,29)(18,24,19,26)(22,32,31,30)(23,27,25,28)

Polycyclic presentation:

2^9|b^2=fgi,c^2=fi,d^2=i,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,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

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6486859);