Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30428 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#30430 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2725 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)(5,9)(6,8)(7,10)(11,12)(13,18)(14,26)(15,31)(16,21)(17,22)(19,28)(20,24)(23,37)(25,35)(27,33)(29,38)(30,39)(32,34)(36,40)(41,42);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,27)(15,24)(16,32)(17,25)(19,36)(20,31)(21,34)(22,35)(23,29)(26,33)(28,40)(30,41)(37,38)(39,42);(11,14,17,30)(12,19,22,38)(13,23,25,40)(15,29,16,28)(18,33,35,42)(20,37,34,36)(21,39,31,26)(24,41,32,27);(4,8)(5,10)(11,13)(12,20)(14,23)(15,32)(16,24)(17,25)(18,31)(19,37)(21,35)(22,34)(26,33)(27,29)(28,41)(30,40)(36,38)(39,42)74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7407928);