Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#31053 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2650 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)(11,12,15,20,16,21,31,38)(13,17,23,19,25,34,24,33)(14,26,28,36,29,37,30,18)(22,39,40,42,41,35,27,32);(3,5)(4,7)(6,10)(8,9)(11,13)(12,17)(14,27)(15,23)(16,25)(18,35)(19,20)(21,34)(22,28)(24,31)(26,32)(29,40)(30,41)(33,38)(36,39)(37,42);(11,14,16,29)(12,18,21,36)(13,22,25,41)(15,30,31,28)(17,32,34,42)(19,35,33,39)(20,37,38,26)(23,27,24,40);(4,8)(5,10)(11,13)(12,19)(14,22)(15,24)(16,25)(17,20)(18,35)(21,33)(23,31)(26,42)(27,28)(29,41)(30,40)(32,37)(34,38)(36,39)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410450);