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×#54049 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2678 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5832 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)(5,9)(6,7)(8,10)(12,15)(13,17)(19,20,23,24)(21,25,27,31)(22,26,29,32)(28,33,30,34);(4,7)(8,9)(11,12,14,15)(13,18,17,16)(20,24)(21,27)(22,29)(33,34);(11,12,14,15)(13,16,17,18)(19,21)(20,25)(22,30)(23,27)(24,31)(26,34)(28,29)(32,33);(3,4)(5,8)(6,7)(9,10)(11,12)(13,16)(14,15)(17,18)(19,22)(20,26)(21,28)(23,29)(24,32)(25,33)(27,30)(31,34);(3,5)(4,8)(6,10)(7,9)(11,13)(12,16)(14,17)(15,18)(19,20)(21,25)(22,26)(23,24)(27,31)(28,33)(29,32)(30,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8006106);