Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 132120576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
| direct product | Q8×#401 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2006 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,3,4)(5,8,7,6);(1,5,3,7)(2,6,4,8);(11,15)(13,16)(18,21)(22,23)(25,26,28,30)(27,31,32,29);(9,10,12,14)(11,16,15,13)(17,18,20,21)(19,23,24,22)(25,26)(27,31)(28,30)(29,32);(9,11)(10,13)(12,15)(14,16)(17,19,20,24)(18,22,21,23)(25,27,28,32)(26,29,30,31)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4214720);