Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C211 |
| direct product | C2×#23425 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
7170 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4)(5,6)(11,12,17,22)(13,23,25,34)(14,28,31,35)(15,21,16,20)(18,27,33,26)(19,29,36,30)(24,38,39,32)(37,40,42,41);(4,6)(7,8)(9,10)(11,13)(12,18)(14,24)(15,27)(16,26)(17,25)(19,32)(20,34)(21,23)(22,33)(28,42)(29,41)(30,40)(31,39)(35,37)(36,38);(8,10)(11,14,17,31)(12,19,22,36)(13,24,25,39)(15,30,16,29)(18,32,33,38)(20,28,21,35)(23,37,34,42)(26,41,27,40);(13,25)(18,33)(23,34)(24,39)(26,27)(32,38)(37,42)(40,41)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6338956);