Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C28 |
| direct product | C2×#12683 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C8 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×C22⋊C8 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C8 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
3302 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,11,16,12,17,26,34,13,18,27,35,10,15,24,32)(9,19,21,36,20,30,37,33,22,31,25,38,23,14,28,29);(7,9)(8,14)(10,23)(11,25)(12,20)(13,22)(15,31)(16,33)(17,19)(18,30)(21,27)(24,37)(26,28)(29,35)(32,36)(34,38);(4,6)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400535);