Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#57869 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2714 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4826 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 42 points:
(1,2);(4,8)(6,9)(11,12,15,16)(13,19,20,17)(14,22,23,18)(21,24,26,25)(28,29)(30,37)(31,39)(40,41);(3,4)(5,9)(6,10)(7,8)(12,16)(13,20)(18,22)(21,26)(27,28,32,29)(30,35,37,33)(31,36,39,34)(38,40,42,41);(4,8)(5,10)(11,12)(13,17)(14,18)(15,16)(19,20)(21,24)(22,23)(25,26)(27,29,32,28)(30,33,37,35)(31,36,39,34)(38,40,42,41);(3,5)(4,6)(7,10)(8,9)(11,13,15,20)(12,17,16,19)(14,21,23,26)(18,24,22,25)(27,30,32,37)(28,33,29,35)(31,38,39,42)(34,40,36,41);(3,6)(4,5)(7,9)(8,10)(11,14,15,23)(12,18,16,22)(13,21,20,26)(17,24,19,25)(27,31,32,39)(28,34,29,36)(30,38,37,42)(33,40,35,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 8008913);