#6571695

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533141
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	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

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, 6478891);