Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33913 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C2×#3339 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1342 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);(3,4,7,6)(5,9,10,8)(11,12)(13,22)(14,19)(15,18)(16,29)(17,23)(20,28)(21,27)(24,30)(25,32)(26,34)(31,40)(33,38)(35,42)(36,41)(37,39);(3,5)(4,8)(6,9)(7,10)(11,13)(12,17)(14,23)(15,24)(16,25)(18,33)(19,22)(20,31)(21,32)(26,38)(27,39)(28,41)(29,37)(30,34)(35,36)(40,42);(3,6,7,4)(5,9,10,8)(12,18)(13,23)(15,28)(16,27)(17,30)(19,34)(20,21)(22,33)(24,40)(26,42)(29,35)(31,37)(32,36)(38,41);(5,10)(8,9)(13,23)(17,22)(24,38)(25,39)(30,33)(31,36)(32,37)(40,41)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530993);