#6572628

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×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3021 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)(19,23)(20,28,31,24)(21,26)(22,25)(27,32,29,30);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,19,22,26)(18,23,25,21)(20,29,31,27)(24,30,28,32);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20,22,31)(18,24,25,28)(19,27,26,29)(21,30,23,32);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(18,25)(21,23)(24,28)(30,32);(4,12)(7,14)(10,16)(13,15)(17,21,22,23)(18,26,25,19)(20,30,31,32)(24,29,28,27)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=h,e^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,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, 6479824);