Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#37018 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4746 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);(4,7)(8,10)(12,16)(14,21)(17,22)(20,25)(27,28)(29,32)(30,35)(31,34)(33,38)(36,39)(37,41)(40,42);(12,16)(13,19)(18,23)(20,25)(28,29)(33,35)(34,36)(40,41);(3,4,6,7)(5,8,9,10)(11,12)(13,17)(14,18)(15,16)(19,22)(20,24)(21,23)(25,26)(27,29,32,28)(30,35,38,33)(31,36,39,34)(37,41,42,40);(3,5,6,9)(4,8,7,10)(11,13,15,19)(12,17,16,22)(14,20,21,25)(18,24,23,26)(27,30)(28,33)(29,35)(31,37)(32,38)(34,40)(36,41)(39,42);(11,14)(12,18)(13,20)(15,21)(16,23)(17,24)(19,25)(22,26)(27,31,32,39)(28,34,29,36)(30,37,38,42)(33,40,35,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7541604);