Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7262 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C23×C4 |
| direct product | C2×#7279 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #340 |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C24 | #102 |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
534 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 42 points:
(1,2);(3,4)(5,7)(6,9)(8,10)(11,12)(13,23,25,37)(14,20)(15,30)(16,32)(17,22)(18,24,33,38)(19,28)(21,31)(26,41,39,35)(27,42,40,34)(29,36);(4,7)(5,10)(8,9)(11,13,15,26,31,40,29,38,17,25,28,39,16,27,14,24)(12,18,20,34,32,41,19,23,22,33,36,42,21,35,30,37)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34913);