#6529164

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

3119 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)(19,22)(20,23)(21,24)(25,31)(26,29)(27,30)(28,32);(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,19)(18,22)(20,25)(21,26)(23,28)(24,29)(27,32)(30,31);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,32)(29,31);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,30)(31,32);(2,8)(6,9)(7,16)(14,15)(18,24)(22,29)(23,30)(28,31)

Polycyclic presentation:

2^9|c^2=h,d^2=h,b^a=bf,c^a=cg,e^a=ei,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, 6436360);