Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C212 |
| direct product | C2×#41911 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C25 |
| commutator, Frattini, cosocle | C23 | C26 |
| prime radical | C2×#392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
5050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 6 generators on 34 points:
(1,2);(5,9)(8,10)(12,15)(13,17)(19,20,24,23)(21,25,29,28)(22,30,32,26)(27,34,31,33);(3,4,7,6)(5,8,9,10)(11,12,14,15)(13,16,17,18)(20,23)(25,28)(26,30)(33,34);(3,5,7,9)(4,8,6,10)(11,12)(13,16)(14,15)(17,18)(19,21,24,29)(20,25,23,28)(22,31,32,27)(26,34,30,33);(3,4,7,6)(5,8,9,10)(11,13,14,17)(12,16,15,18)(19,22)(20,26)(21,27)(23,30)(24,32)(25,33)(28,34)(29,31);(3,6,7,4)(5,10,9,8)(19,23,24,20)(21,28,29,25)(22,30,32,26)(27,34,31,33)104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7547165);