#6585851

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1463 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,c^2=gi,d^2=f,e^2=i,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6493047);