#6349873

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349830
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	#62835	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2770 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)(3,12)(4,9)(5,10)(6,15)(7,11)(8,14)(13,16)(17,18,23,26)(19,29,25,32)(20,31,24,30)(21,27,22,28);(1,3,7,4)(2,8,11,16)(5,14,6,13)(9,15,12,10)(17,19,23,25)(18,24,26,20)(21,30,22,31)(27,29,28,32);(1,4,7,3)(2,9,11,12)(5,14,6,13)(8,15,16,10)(17,20,23,24)(18,25,26,19)(21,29,22,32)(27,30,28,31);(1,5,7,6)(2,10,11,15)(3,13,4,14)(8,12,16,9)(18,26)(19,25)(27,28)(30,31);(1,5,7,6)(2,10,11,15)(3,14,4,13)(8,9,16,12)(17,21,23,22)(18,27,26,28)(19,30,25,31)(20,32,24,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6257069);