Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29736 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
| direct product | C2×#30192 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#2619 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C22×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
2330 conjugacy classes of subgroups.
| minimal degree | μ(G) = 34 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 34 points:
(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20,24,26,23,27,22,28)(21,25,29,34,31,32,30,33);(4,7)(8,10)(11,13,14,17)(12,15,16,18)(19,21,22,30,23,31,24,29)(20,25,28,33,27,32,26,34);(3,5)(4,8)(6,9)(7,10)(12,16)(15,18)(20,26)(21,29)(22,24)(25,32)(27,28)(30,31);(5,9)(8,10)(20,27)(25,32)(26,28)(33,34)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6340919);