#6350543

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

2850 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)(3,10)(4,9)(5,13)(6,8)(7,11)(12,16)(14,15)(18,22)(20,27)(23,29)(24,28)(26,31)(30,32);(1,3,6,12)(2,7,8,15)(4,11,5,14)(9,16,13,10)(20,27)(22,28)(26,31)(29,32);(17,18)(19,23)(20,28)(21,24)(22,27)(25,30)(26,32)(29,31);(2,8)(7,15)(9,13)(10,16)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,31)(28,32);(1,4,6,5)(2,9,8,13)(3,11,12,14)(7,16,15,10)(19,25)(23,30)(26,31)(29,32)

Polycyclic presentation:

2^9|b^2=h,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, 6257739);