Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24468 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#25619 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3512 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3384 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,9)(6,7)(8,10)(11,12)(13,22)(14,26)(15,20)(16,30)(17,21)(18,24)(19,29)(23,40)(25,34)(27,33)(28,38)(31,35)(32,37)(36,41)(39,42);(4,7)(8,9)(11,13)(12,18)(14,27)(15,24)(16,25)(17,31)(19,36)(20,22)(21,34)(23,29)(26,39)(28,42)(30,35)(32,41)(33,38)(37,40);(11,14,16,29,15,28,17,32)(12,19,21,26,20,37,30,38)(13,23,25,42,24,41,31,27)(18,33,34,36,22,39,35,40);(3,5,6,10)(4,8,7,9)(12,20)(14,28)(18,22)(21,30)(23,41)(27,42)(29,32)(34,35)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6356381);