Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #52308 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#52940 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#2825 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4842 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);(3,4)(5,8)(6,10)(7,9)(14,22)(18,23)(20,26)(24,25)(27,28)(29,33)(30,32)(31,38)(34,39)(35,41)(36,37)(40,42);(3,5,7,10)(4,8,9,6)(12,16)(17,19)(18,23)(24,25)(28,30)(33,37)(34,38)(40,41);(3,5)(4,6)(7,10)(8,9)(11,12,15,16)(13,19,21,17)(14,18,22,23)(20,25,26,24)(27,29)(28,33)(30,37)(31,35)(32,36)(34,40)(38,41)(39,42);(3,6)(4,5)(7,8)(9,10)(11,13,15,21)(12,17,16,19)(14,20,22,26)(18,24,23,25)(27,30,32,28)(29,33,36,37)(31,38,39,34)(35,40,42,41);(11,14)(12,18)(13,20)(15,22)(16,23)(17,24)(19,25)(21,26)(27,31,32,39)(28,34,30,38)(29,35,36,42)(33,40,37,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999924);