#6355620

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355609
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

1672 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,22,25)(19,21,24,31)(20,27,30,26)(23,28,32,29);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,15)(12,16)(17,19,21,25,22,24,31,18)(20,26,28,32,30,27,29,23);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20)(18,23)(19,26)(21,28)(22,30)(24,27)(25,32)(29,31);(3,10)(6,14)(11,16)(13,15)(17,21,22,31)(18,24,25,19)(20,28,30,29)(23,27,32,26);(2,8)(6,14)(7,9)(13,15)(17,21,22,31)(18,19,25,24)(20,29,30,28)(23,27,32,26)

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=h,e^2=h,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^c=eh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262816);