Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#54091 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4940 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(4,8)(6,10)(11,12,16,17)(13,15,14,18)(19,20)(21,27)(22,24)(23,30)(25,29)(26,31)(28,32)(33,34);(3,4,7,8)(5,10,9,6)(12,17)(13,14)(20,22)(23,31)(25,27)(28,34);(4,8)(5,9)(11,13,16,14)(12,18,17,15)(19,21,24,29)(20,25,22,27)(23,28,31,34)(26,32,30,33);(3,5,7,9)(4,6,8,10)(11,14,16,13)(12,15,17,18)(19,22,24,20)(21,25,29,27)(23,30,31,26)(28,32,34,33);(3,6)(4,9)(5,8)(7,10)(11,15)(12,13)(14,17)(16,18)(19,23)(20,26)(21,28)(22,30)(24,31)(25,32)(27,33)(29,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006802);