#6354413

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354359
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

1623 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)(4,7)(8,12)(9,13)(11,14)(15,16)(17,18)(19,26)(20,25)(21,29)(22,24)(23,27)(28,32)(30,31);(1,2,7,12,5,8,4,6)(3,9,14,16,10,15,11,13);(2,8)(6,12)(9,15)(13,16)(17,19,22,28)(18,23,24,31)(20,27,21,30)(25,32,29,26);(1,3,5,10)(2,9,8,15)(4,11,7,14)(6,13,12,16)(18,24)(23,31)(25,29)(26,32);(3,10)(9,15)(11,14)(13,16)(17,20,22,21)(18,25,24,29)(19,27,28,30)(23,32,31,26)

Polycyclic presentation:

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

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, 6261609);