#6359135

order	512
nilpotency class	3
isoclinism	isoclinic class of #6359112
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

1813 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)(8,9)(10,15)(11,16)(12,14)(17,18)(19,22)(20,23)(21,25)(24,28)(26,29)(27,31)(30,32);(1,2,6,10)(3,11,13,8)(4,7,5,15)(9,14,16,12)(17,19,21,27)(18,22,25,31)(20,26,28,32)(23,29,24,30);(1,3)(2,8)(4,14)(5,12)(6,13)(7,9)(10,11)(15,16)(18,23)(20,28)(22,29)(24,25)(26,32)(30,31);(1,3)(2,9)(4,12)(5,14)(6,13)(7,8)(10,16)(11,15);(1,4,6,5)(2,7,10,15)(3,12,13,14)(8,16,11,9)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266331);