Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
| direct product | C22×#3172 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator, Frattini, cosocle | C2×C4 | C26 |
| socle | C24 | C22×D4 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
7234 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(5,6)(7,10)(8,9)(11,12)(13,14,17,20)(15,21,22,28)(16,24,25,19)(18,26,27,23);(6,9)(10,12)(15,22)(18,27)(21,28)(23,26);(13,15,16,23,17,22,25,26)(14,18,19,21,20,27,24,28);(5,7,8,11)(6,10,9,12)128 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 10482489);