Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C210×C42 |
| direct product | Q8×H(3,𝔽4) |
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C26 |
| prime radical | C2×H(3,𝔽4) | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5913 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2,3,4)(5,8,7,6);(1,5,3,7)(2,6,4,8);(9,10,13,16)(11,18,19,15)(12,20,22,24)(14,17,23,21);(9,11,13,19)(10,15,16,18)(12,21,22,17)(14,20,23,24);(9,12)(10,11)(13,22)(14,15)(16,19)(17,20)(18,23)(21,24);(9,11)(10,12)(13,19)(14,20)(15,17)(16,22)(18,21)(23,24)95 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7998955);