Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18346 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#18362 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#534 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
2948 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4,7,10)(5,11,12,18)(6,14,16,9)(8,15,17,13)(20,25)(22,23)(26,33)(27,28)(29,31)(32,34);(3,5)(4,8)(6,15)(7,12)(9,18)(10,17)(11,14)(13,16)(19,20)(21,26)(22,25)(23,27)(24,28)(29,32)(30,34)(31,33);(19,21)(20,26)(22,29)(23,31)(24,30)(25,32)(27,33)(28,34);(5,12)(8,17)(11,18)(13,15)(19,22,24,23)(20,27,28,25)(21,29,30,31)(26,33,34,32)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254120);