Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8453 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
| direct product | C2×#8462 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #379 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C2×#3485 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3494 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4)(5,9,10,7)(6,8)(11,12,16,21,17,22,15,20)(13,23,27,18,26,33,25,32)(14,19,28,34,29,35,30,36)(24,38,39,42,40,37,41,31);(3,5)(4,7)(6,10)(8,9)(11,13)(12,18)(14,24)(15,27)(16,25)(17,26)(19,31)(20,33)(21,23)(22,32)(28,41)(29,40)(30,39)(34,37)(35,42)(36,38);(11,14)(12,19)(13,24)(15,30)(16,28)(17,29)(18,31)(20,36)(21,34)(22,35)(23,37)(25,41)(26,40)(27,39)(32,42)(33,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 94121);