Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C25 |
| commutator, Frattini, socle, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4458 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,6)(3,7,8,5)(9,10,14,12)(11,17,18,15)(13,16,21,20)(19,23,24,22)(25,26,29,31)(27,33,34,30)(28,36,38,32)(35,39,37,40);(1,2)(3,5)(4,6)(7,8)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,27)(26,30)(28,37)(29,34)(31,33)(32,40)(35,38)(36,39);(1,3,4,8)(2,5,6,7)(9,12,14,10)(11,17,18,15)(13,20,21,16)(19,23,24,22)(26,31)(27,34)(32,36)(35,37);(2,6)(3,8)(10,12)(15,17)(16,20)(22,23)(25,28,29,38)(26,32,31,36)(27,35,34,37)(30,39,33,40);(2,6)(5,7)(11,18)(15,17)(19,24)(22,23)(26,31)(30,33)(32,36)(39,40);(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)Polycyclic presentation:
2^9|a^2=g,c^2=gi,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8012107);