Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#53387 |
| 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 | |
5126 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);(5,9)(8,10)(12,15)(16,18)(19,20,23,26)(21,24,29,27)(22,30,31,25)(28,33,34,32);(4,7)(8,10)(13,17)(16,18)(19,20)(21,27)(22,25)(23,26)(24,29)(28,33)(30,31)(32,34);(3,4)(5,8)(6,7)(9,10)(11,12,14,15)(13,16,17,18);(3,5,6,9)(4,8,7,10)(19,21)(20,24)(22,28)(23,29)(25,32)(26,27)(30,33)(31,34);(3,4)(5,8)(6,7)(9,10)(11,13)(12,16)(14,17)(15,18)(19,22,23,31)(20,25,26,30)(21,28,29,34)(24,32,27,33)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8013080);