Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29748 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#29886 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,9)(6,10)(7,8)(11,17)(12,15)(13,18)(14,16)(19,20,22,27,24,28,23,26)(21,25,30,33,31,32,29,34);(4,8)(9,15)(10,16)(17,18)(19,21)(20,25)(22,30)(23,29)(24,31)(26,34)(27,33)(28,32);(3,5,7,12)(4,9,8,15)(6,13,14,11)(10,18,16,17)(20,26)(21,29)(22,23)(25,32)(27,28)(30,31);(3,6,7,14)(4,10,8,16)(5,11,12,13)(9,17,15,18)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6343406);