#6349281

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349215
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
prime radical	C₂×#3506	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

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, 6256477);