#6614517

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1403 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)(3,8)(4,7)(5,6)(17,18,22,26,23,24,21,25)(19,27,28,30,29,31,32,20);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19)(18,20)(21,28)(22,32)(23,29)(24,30)(25,27)(26,31);(1,3,5,4)(2,7,6,8)(10,13)(11,15)(17,20)(18,19)(21,31)(22,27)(23,30)(24,29)(25,32)(26,28);(1,4,5,3)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(18,24)(20,30)(25,26)(27,31);(10,13)(11,15)(17,21,23,22)(18,25,24,26)(19,28,29,32)(20,27,30,31)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6521713);