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×#56694 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(3,4,7,8)(5,6,9,10)(11,12)(13,16)(14,15)(17,18)(19,20)(21,25)(22,30)(23,24)(26,31)(27,28)(29,34)(32,33);(4,8)(5,9)(11,13,15,18)(12,16,14,17)(20,24)(21,27)(22,31)(32,34);(12,14)(16,17)(19,21,23,27)(20,25,24,28)(22,29,31,33)(26,32,30,34);(3,5)(4,6)(7,9)(8,10)(11,14,15,12)(13,17,18,16)(19,20)(21,28)(22,26)(23,24)(25,27)(29,34)(30,31)(32,33);(3,6)(4,5)(7,10)(8,9)(11,13)(12,16)(14,17)(15,18)(19,22)(20,26)(21,29)(23,31)(24,30)(25,32)(27,33)(28,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006838);