Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
| direct product | C2×#7125 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #327 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| socle | C22 | #1271 |
| prime radical | C24 | #104 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
408 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 34 points:
(1,2);(3,4)(5,13,15,26,16,30,21,33)(6,17,19,10)(7,12)(8,11)(9,14,23,31,24,32,25,18)(20,29,34,27)(22,28);(3,5)(4,9,11,24)(6,18)(7,21)(8,16)(10,26,17,33)(12,23,28,25)(13,29,30,27)(14,20)(15,22)(19,31)(32,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31291);