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×#33947 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| prime radical | C22×D16 | C22 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1318 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)(13,17)(14,18)(15,28)(16,21)(19,25)(20,26)(22,29)(23,38)(24,32)(27,34)(30,35)(31,36)(33,39)(37,41)(40,42);(3,5,7,10)(4,8,6,9)(11,13,16,24)(12,17,21,32)(14,22,27,37)(15,23,26,36)(18,29,34,41)(19,30,33,40)(20,31,28,38)(25,35,39,42);(12,18)(14,25)(15,26)(17,29)(19,20)(21,34)(22,35)(23,36)(27,39)(28,33)(30,31)(32,41)(37,42)(38,40);(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, 7531108);