#6458480

order	512
nilpotency class	3
isoclinism	isoclinic class of #6447056
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂	C₂×#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1139 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=g,c^2=fg,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,h^b=hi,e^c=ei

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, 6365676);