#6350592

order	512
nilpotency class	3
isoclinism	isoclinic class of #6350525
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2950 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257788);