Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#31205 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2358 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(3,4,5,6)(7,8,11,14)(9,10,13,12)(17,24)(21,29)(25,30)(26,28);(7,9,11,13)(8,12,14,10)(15,16)(17,25)(18,22)(19,23)(20,27)(21,26)(24,30)(28,29);(4,6)(7,10,11,12)(8,13,14,9)(15,17)(16,21)(18,26)(19,24)(20,28)(22,30)(23,29)(25,27);(4,6)(8,14)(9,13)(15,18,19,20)(16,22,23,27)(17,26,24,28)(21,30,29,25)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410937);