#6572572

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533135
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
prime radical	#3459	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2899 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6479768);