#6347298

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347252
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	C₂²×C₄	C₂⁵
prime radical	#62812	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

3855 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:

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

Polycyclic presentation:

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

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, 6254494);