#6349756

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349684
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	#60900	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2633 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)(3,11)(4,7)(5,10)(6,9)(8,14)(12,13)(15,16)(17,18)(19,28)(20,30)(21,26)(22,32)(23,27)(24,29)(25,31);(1,3,7,15)(2,8,4,12)(5,14,6,13)(9,11,10,16)(17,19,23,25)(18,24,27,20)(21,29,22,30)(26,31,32,28);(1,4,7,2)(3,12,15,8)(5,10,6,9)(11,14,16,13)(17,20,23,24)(18,25,27,19)(21,31,22,28)(26,29,32,30);(2,4)(8,12)(9,10)(11,16)(17,21,23,22)(18,26,27,32)(19,29,25,30)(20,28,24,31);(1,5,7,6)(2,9,4,10)(3,13,15,14)(8,16,12,11)(18,27)(19,25)(26,32)(29,30)

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=i,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=eh,f^a=fh,g^a=gi,e^b=eh,f^b=fh,g^b=gi,d^c=dh,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, 6256952);