Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1795 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C28 |
| direct product | C2×#8324 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C23×#41 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3514 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4,5,7,6,8,9,10)(11,12)(13,20,22,17)(14,16)(15,19)(18,23)(21,25,26,24);(12,16)(14,23)(17,24)(18,19)(20,25)(21,26);(11,13,15,22)(12,17,19,20)(14,21,23,26)(16,24,18,25)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 92165);