Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#9536 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle, prime radical | C24 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
896 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 26 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12,13,15)(14,16,18,17)(19,20,24,21,23,26,22,25);(4,7)(8,10)(12,15)(16,17)(19,21,24,26,23,25,22,20);(3,5,6,9)(4,8,7,10)(12,16)(14,18)(15,17)(19,22,23,24)(20,25,26,21)152 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 109379);