#6351983

order	512
nilpotency class	3
isoclinism	isoclinic class of #6351885
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

2827 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)(3,13)(4,14)(5,10)(6,11)(7,12)(8,15)(9,16)(17,18)(19,26)(20,23)(21,25)(22,30)(24,28)(27,31)(29,32);(1,3,7,9)(2,8,12,4)(5,14,6,15)(10,13,11,16)(17,19)(18,24)(20,27)(21,28)(22,31)(23,29)(25,32)(26,30);(1,4,7,8)(2,9,12,3)(5,16,6,13)(10,15,11,14)(17,18,23,20)(19,24,29,27)(21,25,22,30)(26,28,32,31);(1,2)(3,8)(4,9)(5,11)(6,10)(7,12)(13,15)(14,16)(17,20,23,18)(19,27,29,24)(21,25,22,30)(26,28,32,31);(1,5,7,6)(2,10,12,11)(3,14,9,15)(4,16,8,13)(17,21,23,22)(18,25,20,30)(19,28,29,31)(24,32,27,26)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6259179);