Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#46176 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
4584 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)(11,12,16,13)(14,19,22,17)(15,20,23,18)(21,24,26,25)(27,28)(29,33)(30,37)(31,32)(34,38)(35,41)(36,39)(40,42);(12,13)(15,23)(17,19)(21,26)(28,32)(33,39)(34,37)(40,41);(3,4,6,7)(5,8,9,10)(11,13,16,12)(14,19,22,17)(15,20,23,18)(21,25,26,24)(27,29,31,36)(28,33,32,39)(30,35,38,42)(34,40,37,41);(3,5,6,9)(4,8,7,10)(11,14,16,22)(12,17,13,19)(15,21,23,26)(18,24,20,25)(27,28)(29,33)(30,34)(31,32)(35,40)(36,39)(37,38)(41,42);(11,15,16,23)(12,18,13,20)(14,21,22,26)(17,24,19,25)(27,30,31,38)(28,34,32,37)(29,35,36,42)(33,40,39,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7609988);