Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 37748736 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
| direct product | Q8×#395 |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
4058 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(5,8,7,6);(1,5,3,7)(2,6,4,8);(14,16)(17,18,21,20)(19,23,24,22);(10,12)(14,16)(17,19,21,24)(18,22,20,23);(9,10)(11,12)(13,14)(15,16)(17,20,21,18)(19,22,24,23)110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4214646);