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×#2833 |
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C25 | C24 |
| commutator, Frattini, cosocle | C23 | C26 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C25 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5930 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 36 points:
(1,2);(3,4);(6,9)(10,12)(13,14,17,18)(15,19,20,16)(22,26)(27,29)(28,32)(35,36);(7,11)(10,12)(15,20)(16,19)(21,22,25,26)(23,29,31,27)(24,32,34,28)(30,35,33,36);(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,20)(18,19)(21,23,25,31)(22,27,26,29)(24,33,34,30)(28,36,32,35);(5,7)(6,10)(8,11)(9,12)(13,16)(14,15)(17,19)(18,20)(21,24,25,34)(22,28,26,32)(23,30,31,33)(27,35,29,36)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7540578);