#6355595

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355552
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1676 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262791);