Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#28811 |
| 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 ◁ C2×C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2002 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);(7,8,10,12)(9,13,14,11)(15,16,20,22)(17,25,18,28)(19,30,26,24)(21,29,23,27);(3,4,5,6)(15,17,19,27,20,18,26,29)(16,21,24,25,22,23,30,28);(4,6)(7,9,10,14)(8,11,12,13)(16,22)(21,23)(24,30)(25,28);(15,18,19,29,20,17,26,27)(16,23,24,28,22,21,30,25)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372933);