#6349012

order	512
nilpotency class	3
isoclinism	isoclinic class of #6348967
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×#3480	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2954 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,g^a=gi,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, 6256208);