#6351450

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2732 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)(7,15)(8,12)(9,13)(14,16)(17,18)(19,26)(20,24)(21,29)(22,25)(23,27)(28,31)(30,32);(1,2)(3,7)(4,12)(5,8)(6,11)(9,14)(10,16)(13,15)(17,18)(19,23)(20,24)(21,29)(22,25)(26,30)(27,31)(28,32);(1,3,5,10)(2,7,8,16)(4,13,11,9)(6,14,12,15)(17,19)(18,23)(20,27)(21,30)(22,28)(24,31)(25,32)(26,29);(2,8)(6,12)(7,16)(14,15);(17,20,22,21)(18,24,25,29)(19,27,28,30)(23,31,32,26)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6258646);