#6587856

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1691 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)(12,13)(15,16)(17,18)(19,24)(20,25)(21,30)(22,26)(23,27)(28,32)(29,31);(2,6)(7,8)(17,19,23,29)(18,24,27,31)(20,21,28,22)(25,30,32,26);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(19,20)(21,22)(24,25)(26,30)(28,29)(31,32);(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,21)(22,29)(23,28)(24,30)(26,31)(27,32);(1,4,5,3)(2,8,6,7)(17,21)(18,26)(20,28)(22,23)(24,31)(27,30)

Polycyclic presentation:

2^9|b^2=i,c^2=f,d^2=g,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=ehi,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, 6495052);