Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23886 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#24930 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3013 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2673 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)(5,7)(6,8)(9,10)(11,12)(13,22)(14,27)(15,17)(16,21)(18,26)(19,28)(20,31)(23,32)(24,39)(25,34)(29,36)(30,35)(33,41)(37,38)(40,42);(3,5,6,9)(4,7,8,10)(12,17)(13,23)(15,31)(19,27)(20,21)(24,38)(25,26)(28,30)(32,34)(35,36)(39,42)(40,41);(4,8)(7,10)(11,13)(12,18)(14,24)(15,25)(16,26)(17,32)(19,33)(20,34)(21,22)(23,31)(27,42)(28,38)(29,40)(30,41)(35,37)(36,39);(11,14,16,29)(12,19,21,35)(13,24,26,40)(15,30,31,28)(17,27,20,36)(18,33,22,37)(23,38,25,41)(32,42,34,39)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6345573);