#6534943

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 ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3006 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:

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

Polycyclic presentation:

2^9|d^2=h,e^2=hi,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, 6442139);