Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8610 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
| direct product | C2×#8625 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #380 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C24 | C2×C4⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
882 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,7,10,8,11,6,16)(5,12,15,17,13,18,14,9)(19,20,22,26,23,25,31,33)(21,27,28,34,29,24,30,32);(3,5,6,14,8,13,7,15)(4,9,10,12,11,17,16,18);(5,13)(9,17)(12,18)(14,15)(19,21,23,29)(20,24,25,27)(22,30,31,28)(26,34,33,32)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 96109);