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×#52868 |
| 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 | |
4534 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);(4,9)(5,7)(12,16)(13,19)(14,22)(24,26)(27,28)(29,33)(30,32)(31,38)(34,39)(35,41)(36,37)(40,42);(3,4,8,9)(5,6,7,10)(28,30)(33,37)(34,38)(40,41);(3,5,8,7)(4,10,9,6)(11,12)(13,20)(14,18)(15,16)(17,19)(21,26)(22,23)(24,25)(27,29,32,36)(28,33,30,37)(31,35,39,42)(34,40,38,41);(3,6,8,10)(4,5,9,7)(11,13)(12,17)(14,21)(15,19)(16,20)(18,24)(22,25)(23,26)(27,30,32,28)(29,33,36,37)(31,38,39,34)(35,40,42,41);(3,7,8,5)(4,10,9,6)(11,14,15,22)(12,18,16,23)(13,21,19,25)(17,24,20,26)(27,31)(28,34)(29,35)(30,38)(32,39)(33,40)(36,42)(37,41)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7999757);