Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11911 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #329 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | #97 | C24 |
| socle | C22 | C2×#329 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
832 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,12,6,10,19,27,8,13,22,31,20,11,28,29)(5,14,16,32,17,9,23,24,18,25,33,30,15,26,21,34);(3,5)(4,9)(6,15)(7,21)(8,18)(10,14)(11,25)(12,30)(13,26)(16,28)(17,20)(19,33)(22,23)(24,27)(29,34)(31,32);(3,6,8,20)(4,10,13,11)(5,15,18,17)(7,19,22,28)(9,14,26,25)(12,27,31,29)(16,21,33,23)(24,32,34,30)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 390204);