Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1619 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
| direct product | C22×#1649 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | #120 |
| commutator, Frattini, cosocle | C42 | C25 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
4182 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(1,2);(3,4);(5,6)(7,9)(8,11)(10,12)(13,14,19,23)(15,24,16,27)(17,22,18,21)(20,26,28,25);(6,9)(7,12)(10,11)(13,15,19,16)(14,20,23,28)(17,26,18,25)(21,27,22,24);(13,16,19,15)(14,20,23,28)(17,25,18,26)(21,27,22,24)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6255237);