Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24412 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#25921 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3120 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 26 points:
(1,2);(3,4,7,10)(5,9,6,8)(11,12)(13,18)(14,21)(15,19)(16,25)(17,20)(22,23)(24,26);(3,5,7,6)(4,8,10,9)(11,13)(12,18)(14,19)(15,21)(16,26)(17,22)(20,23)(24,25);(3,5,7,6)(4,9,10,8)(11,14,16,18,17,24,15,23)(12,13,19,21,20,22,25,26);(3,6,7,5)(4,9,10,8)(11,15,17,16)(12,19,20,25)(13,21,22,26)(14,23,24,18)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6354930);