Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24805 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#28499 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3153 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3072 conjugacy classes of subgroups.
| minimal degree | μ(G) = 38 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 38 points:
(1,2);(7,8,13,18,14,19,12,28)(9,20,22,31,23,17,11,15)(10,24,26,34,27,33,29,16)(21,30,25,32,36,38,37,35);(3,4,5,6)(7,9,12,11,14,23,13,22)(8,15,18,20,19,31,28,17)(10,21,26,25,27,36,29,37)(16,30,33,35,34,38,24,32);(4,6)(7,10)(8,16)(9,21)(11,25)(12,26)(13,29)(14,27)(15,30)(17,32)(18,33)(19,34)(20,35)(22,37)(23,36)(24,28)(31,38);(7,11,12,23,14,22,13,9)(8,17,18,15,19,20,28,31)(10,25,26,36,27,37,29,21)(16,32,33,30,34,35,24,38)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372787);