Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | C2×#33950 |
| 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 | |
1242 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)(11,12,17,21)(13,18,16,31)(14,24,26,41)(15,27,30,40)(19,34,36,25)(20,23,38,29)(22,39,32,37)(28,33,42,35);(3,5,7,10)(4,8,6,9)(11,13,17,16)(12,18,21,31)(14,22,26,32)(15,23,30,29)(19,33,36,35)(20,27,38,40)(24,39,41,37)(25,28,34,42);(11,14,17,26)(12,19,21,36)(13,22,16,32)(15,28,30,42)(18,33,31,35)(20,37,38,39)(23,34,29,25)(24,40,41,27);(3,6,7,4)(5,8,10,9)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7531111);