Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | C2×#33849 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C25 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×D4 ◁ C2×C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
1990 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(3,4)(5,6)(7,8,14,10)(9,19,21,34)(11,25,28,17)(12,24,31,16)(13,26,30,18)(15,27,20,22)(23,37,32,35)(29,36,38,33);(7,9)(8,15)(10,20)(11,27)(12,29)(13,32)(14,21)(16,34)(17,35)(18,36)(19,24)(22,28)(23,30)(25,37)(26,33)(31,38);(4,6)(7,10,14,8)(9,20,21,15)(11,24,28,16)(12,25,31,17)(13,26,30,18)(19,22,34,27)(23,36,32,33)(29,37,38,35);(7,11)(8,16)(9,21)(10,24)(12,30)(13,31)(14,28)(15,20)(17,26)(18,25)(19,37)(22,38)(27,29)(34,35)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7530794);