#6561005

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535664
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

1364 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=gi,d^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,e^b=ei,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, 6468201);