#6358574

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358426
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

1585 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:

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

Polycyclic presentation:

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

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, 6265770);