#6359006

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358842
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1589 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(18,23)(20,21)(24,31)(25,30)(26,32)(27,28);(17,18)(19,26)(20,30)(21,23)(22,25)(24,28)(27,32)(29,31);(1,3,6,12)(2,7,9,16)(4,11,5,14)(8,15,13,10)(17,18)(19,24)(20,23)(21,30)(22,25)(26,28)(27,31)(29,32);(17,19)(18,24)(20,27)(21,28)(22,29)(23,31)(25,32)(26,30);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,21)(18,23,25,30)(19,27,29,28)(24,31,32,26)

Polycyclic presentation:

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

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, 6266202);