#6349150

order	512
nilpotency class	3
isoclinism	isoclinic class of #6349137
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	#60949	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2907 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(2,6)(3,4)(7,8)(9,10)(11,15)(12,22)(13,25)(14,17)(16,24)(18,28)(19,35)(20,33)(21,30)(23,34)(26,31)(27,32)(29,37)(36,39)(38,40);(1,2)(3,6)(4,8)(5,7)(9,11,14,21)(10,15,17,30)(12,19,23,36)(13,26,28,20)(16,29,32,40)(18,33,25,31)(22,39,34,35)(24,38,27,37);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,19)(13,27)(14,23)(15,29)(17,32)(18,34)(20,38)(21,36)(22,25)(24,28)(26,37)(30,40)(31,39)(33,35);(10,17)(15,30)(16,32)(18,25)(22,34)(29,40)(31,33)(35,39);(2,7)(6,8)

Polycyclic presentation:

2^9|b^2=i,c^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,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

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6256346);