#6529465

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500226
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2791 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=h,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6436661);