Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#25943 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3210 conjugacy classes of subgroups.
| minimal degree | μ(G) = 30 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 30 points:
(1,2);(4,6)(7,8,11,14)(9,13,10,12)(15,16)(17,22)(18,28)(19,23)(20,24)(21,25)(26,29)(27,30);(7,9,11,10)(8,12,14,13)(15,17)(16,22)(18,19)(20,26)(21,27)(23,28)(24,29)(25,30);(3,4,5,6)(7,9,11,10)(8,13,14,12)(15,18,19,17,21,26,20,27)(16,22,24,28,25,30,23,29);(7,10,11,9)(8,13,14,12)(15,19,21,20)(16,23,25,24)(17,18,27,26)(22,28,30,29)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354871);