#6580296

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	#24434	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1794 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,d^2=f,h^2=i,b^a=bf,c^a=cg,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, 6487492);