#6357000

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356901
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 ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1562 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)(4,12)(7,13)(8,14)(9,16)(11,15)(17,18)(19,21)(20,27)(22,25)(23,26)(24,30)(28,31)(29,32);(1,2,4,6,5,7,12,13)(3,9,11,16,10,8,15,14)(17,19,21,18,23,25,22,26)(20,28,29,32,31,27,30,24);(2,7)(6,13)(8,9)(14,16)(17,20,23,31)(18,24,26,32)(19,27,25,28)(21,29,22,30);(1,3,5,10)(2,8,7,9)(4,11,12,15)(6,14,13,16)(17,21,23,22)(18,19,26,25)(20,29,31,30)(24,27,32,28);(3,10)(8,9)(11,15)(14,16)

Polycyclic presentation:

2^9|b^2=fi,c^2=i,d^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=dh,g^c=gi,e^d=ehi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264196);