Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#43629 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5132 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);(3,4)(5,7)(6,8)(9,10)(12,15)(16,18)(19,20,24,22)(21,27,28,25)(23,30,31,26)(29,32,34,33);(3,4,6,8)(5,7,9,10)(21,28)(23,31)(25,27)(26,30);(11,12)(13,16)(14,15)(17,18)(19,21)(20,25)(22,27)(23,29)(24,28)(26,32)(30,33)(31,34);(3,4)(5,7)(6,8)(9,10)(11,13,14,17)(12,16,15,18)(19,22,24,20)(21,25,28,27)(23,30,31,26)(29,32,34,33);(3,5,6,9)(4,7,8,10)(19,23)(20,26)(21,29)(22,30)(24,31)(25,32)(27,33)(28,34)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7574983);