#6477586

order	512
nilpotency class	3
isoclinism	isoclinic class of #6470231
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

3624 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)(9,20)(11,24)(12,26)(16,29)(19,31)(22,23)(25,30)(27,32);(1,2)(3,11)(4,9)(5,8)(6,10)(7,14)(12,25)(13,24)(15,20)(16,23)(17,21)(18,28)(19,26)(22,29)(27,31)(30,32);(1,3,5,14)(2,7,8,11)(4,12,15,26)(6,13,17,28)(9,19,20,25)(10,18,21,24)(16,27,29,32)(22,31,23,30);(2,8)(4,16)(7,11)(9,22)(10,21)(12,27)(15,29)(18,24)(19,31)(20,23)(25,30)(26,32);(1,4)(2,9)(3,12)(5,15)(6,16)(7,19)(8,20)(10,23)(11,25)(13,27)(14,26)(17,29)(18,30)(21,22)(24,31)(28,32)

Polycyclic presentation:

2^9|c^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,g^b=gi,f^c=fi,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, 6384782);