#6590142

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1831 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,16)(13,15)(17,18,21,23)(19,20,24,30)(22,25,31,32)(26,29,27,28);(2,6)(7,8)(9,11,13,12)(10,14,15,16)(17,19,21,24)(18,20,23,30)(22,26,31,27)(25,29,32,28);(1,3,5,4)(2,7,6,8)(10,15)(14,16)(18,23)(19,26)(20,28)(22,31)(24,27)(29,30);(9,12,13,11)(10,16,15,14)(17,20,21,30)(18,24,23,19)(22,29,31,28)(25,27,32,26);(1,4,5,3)(2,8,6,7)(19,26)(20,29)(22,31)(24,27)(25,32)(28,30)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=f,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,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, 6497338);