#6529167

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499892
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 ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3482 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:

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

Polycyclic presentation:

2^9|c^2=h,d^2=h,b^a=bf,c^a=cg,c^b=ch,e^b=eh,g^b=gi,d^c=dh,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, 6436363);