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×#33954 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1248 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 42 points:
(1,2);(3,4,7,6)(5,9,10,8)(11,12,16,20)(13,21,22,39)(14,25,29,40)(15,30,27,19)(17,32,33,23)(18,35,37,28)(24,41,31,36)(26,38,42,34);(3,5,7,10)(4,8,6,9)(13,22)(17,33)(21,39)(23,32)(24,31)(26,42)(34,38)(36,41);(11,13,16,22)(12,17,20,33)(14,26,29,42)(15,31,27,24)(18,36,37,41)(19,38,30,34)(21,40,39,25)(23,35,32,28);(3,6,7,4)(5,8,10,9)86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531116);