#6354089

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354077
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1706 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,d^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,f^a=fh,g^a=gi,e^b=eh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6261285);