Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#31111 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#537 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1798 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)(5,6)(7,9)(8,10)(11,12,16,19,17,22,15,21)(13,18,23,34,26,31,25,33)(14,27,30,20,29,36,28,35)(24,38,39,32,40,42,37,41);(3,5)(4,8)(6,9)(7,10)(11,13)(12,18)(14,24)(15,25)(16,23)(17,26)(19,34)(20,32)(21,33)(22,31)(27,38)(28,37)(29,40)(30,39)(35,41)(36,42);(3,6,7,8)(4,5,9,10)(12,19)(13,23)(15,16)(18,31)(20,27)(21,22)(24,37)(25,26)(28,30)(32,41)(35,36)(39,40);(4,9)(5,10)(11,14)(12,20)(13,24)(15,30)(16,28)(17,29)(18,32)(19,27)(21,36)(22,35)(23,37)(25,39)(26,40)(31,41)(33,42)(34,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7410652);