#6590419

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1734 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)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,22,23)(19,27,29,24)(20,25,30,32)(21,26,31,28);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(18,23)(19,24)(21,26)(25,32)(27,29)(28,31);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,19)(18,24)(20,21)(22,29)(23,27)(25,28)(26,32)(30,31);(9,12,13,11)(10,16,15,14)(17,20,22,30)(18,25,23,32)(19,28,29,26)(21,24,31,27);(1,4,5,3)(2,6,8,7)(17,21)(18,26)(19,20)(22,31)(23,28)(24,32)(25,27)(29,30)

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=fi,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,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, 6497615);