Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12804 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×Q8 | C24 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
1154 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);(4,6)(7,8,12,17,13,18,10,22)(9,14,19,29,21,30,23,37)(11,16,24,32,26,33,25,15)(20,28,35,27,36,38,34,31);(3,4,5,6)(7,9)(8,14)(10,23)(11,20)(12,19)(13,21)(15,31)(16,28)(17,29)(18,30)(22,37)(24,35)(25,34)(26,36)(27,32)(33,38);(7,10,13,12)(8,15,18,32)(9,19,21,23)(11,25,26,24)(14,27,30,31)(16,17,33,22)(20,35,36,34)(28,37,38,29)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400918);