Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1999 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×#2056 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C2×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
5106 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2);(3,4);(5,6)(7,8)(13,14,18,22)(15,20,24,28)(16,25,23,19)(17,27,26,21);(9,10)(11,12)(13,15,18,24)(14,19,22,25)(16,26,23,17)(20,21,28,27);(6,8)(10,12)(13,16,17,15,18,23,26,24)(14,20,21,25,22,28,27,19)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6339148);