#9718195

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,d^2=h,e^2=gi,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9625391);