Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1510 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28×C4 |
| direct product | C22×#1515 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C2×C22⋊C4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C22×C4 | C25 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3346 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 36 points:
(1,2);(3,4);(6,11,13,25)(7,16)(8,20)(12,27,28,36)(14,26,15,21)(18,33)(19,32)(22,23)(24,29,35,30)(31,34);(5,6,9,14)(7,12,19,29)(8,21,20,25)(10,15,17,13)(11,23,26,22)(16,30,32,28)(18,27,34,35)(24,33,36,31);(5,7,17,32)(6,12,13,28)(8,18,23,33)(9,19,10,16)(11,24,25,35)(14,29,15,30)(20,34,22,31)(21,27,26,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250523);