Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C22×#2831 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C26 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
6654 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 28 points:
(1,2);(3,4);(6,9)(10,12)(13,14,17,15)(16,18,20,19)(21,22,25,26)(23,27,28,24);(7,11)(10,12)(14,15)(16,20)(23,28)(24,27);(5,6)(7,10)(8,9)(11,12)(13,15,17,14)(16,18,20,19)(21,23)(22,24)(25,28)(26,27);(5,7)(6,10)(8,11)(9,12)(13,16,17,20)(14,18,15,19)(21,24)(22,23)(25,27)(26,28)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540573);