#6561142

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535648
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 ◁ C₄²×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1430 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=g,d^2=fgi,e^2=i,f^2=i,g^2=i,b^a=bf,c^a=cg,f^a=fi,g^a=gi,c^b=ch,d^b=di,d^c=di,e^d=ehi

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, 6468338);