Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2212 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12822 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #459 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C22 | #1802 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1080 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 34 points:
(1,2);(3,4,9,13,10,15,7,23)(5,14,8,24,18,31,17,11)(6,19,21,34,22,26,20,30)(12,25,27,33,29,32,28,16);(3,5,7,17,10,18,9,8)(4,11,13,14,15,24,23,31)(6,16,20,32,22,33,21,25)(12,26,27,30,29,19,28,34);(3,6)(4,12)(5,16)(7,20)(8,25)(9,21)(10,22)(11,26)(13,27)(14,30)(15,29)(17,32)(18,33)(19,24)(23,28)(31,34)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 400980);