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 | |
4559 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2)(3,7)(4,6)(5,8)(9,10)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,30,32)(27,31,35,38)(28,33,29,37)(34,39,36,40);(1,3)(2,7)(4,8)(5,6)(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(25,27,30,35)(26,31,32,38)(28,36,29,34)(33,40,37,39);(1,4,6,2)(3,7,5,8)(9,10)(11,14)(12,16)(13,15)(17,18)(19,21)(20,22)(23,24)(26,32)(31,38)(33,37)(39,40);(1,3,6,5)(2,7,4,8)(10,15)(14,17)(16,22)(21,23)(25,28,30,29)(26,33,32,37)(27,34,35,36)(31,39,38,40);(1,5,6,3)(2,7,4,8)(11,18)(14,17)(19,24)(21,23)(25,29,30,28)(26,33,32,37)(27,34,35,36)(31,40,38,39);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)Polycyclic presentation:
2^9|a^2=hi,b^2=hi,c^2=i,d^2=h,e^2=h,f^2=g,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, 8011022);