Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41606 |
| 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 | |
5448 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);(4,7)(8,10)(11,12,15,13)(14,17,18,16)(19,20)(21,25)(22,29)(23,24)(26,31)(27,34)(28,32)(30,33);(12,13)(16,17)(20,24)(25,32)(26,29)(33,34);(3,4)(5,8)(6,7)(9,10)(11,13,15,12)(14,17,18,16)(19,21,23,28)(20,25,24,32)(22,30,31,27)(26,34,29,33);(11,14)(12,16)(13,17)(15,18)(19,22)(20,26)(21,27)(23,31)(24,29)(25,33)(28,30)(32,34);(3,5,6,9)(4,8,7,10)(19,20)(21,25)(22,26)(23,24)(27,33)(28,32)(29,31)(30,34)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7546853);