Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C42 |
| direct product | C2×#30553 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2665 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,6,8)(5,9,10,7)(11,12,15,21)(13,23,25,18)(14,27,29,39)(16,32,17,22)(19,31,38,30)(20,26,35,33)(24,36,41,37)(28,40,42,34);(3,5,6,10)(4,7,8,9)(11,13,15,25)(12,18,21,23)(14,28,29,42)(16,26,17,33)(19,36,38,37)(20,22,35,32)(24,30,41,31)(27,34,39,40);(11,14,15,29)(12,19,21,38)(13,24,25,41)(16,31,17,30)(18,34,23,40)(20,37,35,36)(22,39,32,27)(26,42,33,28);(4,8)(5,10)(11,13)(12,20)(14,24)(15,25)(16,33)(17,26)(18,32)(19,37)(21,35)(22,23)(27,34)(28,31)(29,41)(30,42)(36,38)(39,40)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408330);