Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
| direct product | C2×#31049 |
| 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 | |
2660 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)(11,12,15,21,17,23,16,22)(13,18,25,34,26,35,32,20)(14,27,29,38,31,37,30,19)(24,36,28,41,39,42,40,33);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,28)(15,25)(16,32)(17,26)(19,36)(20,22)(21,34)(23,35)(24,30)(27,41)(29,39)(31,40)(33,37)(38,42);(11,14,17,31)(12,19,23,38)(13,24,26,39)(15,30,16,29)(18,33,35,41)(20,36,34,42)(21,37,22,27)(25,40,32,28);(4,8)(5,10)(11,13)(12,20)(14,24)(15,32)(16,25)(17,26)(18,22)(19,36)(21,35)(23,34)(27,33)(28,30)(29,40)(31,39)(37,41)(38,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410445);