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 | |
4568 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 40 points:
(1,2,4,5)(3,7,8,6)(9,10,15,17)(11,19,22,16)(12,13,18,14)(20,21,24,23)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(1,2)(3,7)(4,5)(6,8)(9,11)(10,16)(12,20)(13,23)(14,21)(15,22)(17,19)(18,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,2)(3,6)(4,5)(7,8)(9,12)(10,13)(11,20)(14,17)(15,18)(16,23)(19,21)(22,24)(27,33)(28,35)(30,36)(31,38);(2,5)(6,7)(10,17)(12,18)(16,19)(20,24)(25,28,29,35)(26,31,32,38)(27,34,33,39)(30,37,36,40);(1,3,4,8)(2,6,5,7)(9,13,15,14)(10,18,17,12)(11,21,22,23)(16,20,19,24)(27,33)(30,36)(34,39)(37,40);(9,14,15,13)(10,12,17,18)(11,21,22,23)(16,20,19,24)Polycyclic presentation:
2^9|a^2=gi,d^2=i,e^2=g,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, 8011649);