Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#25306 |
| 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 | |
2152 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)(9,13)(10,11)(12,14)(15,16,18,25,19,24,20,23)(17,26,29,30,28,21,27,22);(3,4)(5,6)(8,11)(12,13)(15,17,18,29,19,28,20,27)(16,21,25,22,24,26,23,30);(4,6)(15,17,19,28)(16,22,24,30)(18,27,20,29)(21,23,26,25);(7,9,10,14)(8,12,11,13)(15,18,19,20)(16,23,24,25)(17,27,28,29)(21,22,26,30)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6355301);