Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1339 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
| direct product | C2×#7126 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #327 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×#97 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C2×#97 ▷ C22×C4 ▷ C2 ▷ C1 | |
460 conjugacy classes of subgroups.
| minimal degree | μ(G) = 26 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 26 points:
(1,2);(4,7,8,9)(11,12,14,19,15,21,13,22,16,20,23,17,25,26,24,18);(3,4)(5,9)(6,8)(7,10)(12,17,20,19)(13,23)(14,24)(15,25)(18,21,22,26)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 31292);