#6587798

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573485
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	#25125	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1613 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)(3,7)(4,8)(5,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,25)(20,23)(21,26)(22,32)(24,27)(28,30)(29,31);(2,6)(7,8)(17,19,23,29)(18,25,20,31)(21,22,30,24)(26,32,28,27);(1,3,5,4)(2,7,6,8)(10,13)(14,16)(17,20,23,18)(19,28,29,26)(21,31,30,25)(22,32,24,27);(9,11,12,15)(10,14,13,16)(17,21)(18,26)(19,22)(20,28)(23,30)(24,29)(25,32)(27,31);(1,4,5,3)(2,8,6,7)(17,22)(18,27)(20,32)(21,30)(23,24)(25,31)

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,h^c=hi,e^d=ehi,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, 6494994);