Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24802 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#28496 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3126 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)(9,17,11,27,22,34,21,15)(10,23,25,33,26,32,29,16)(20,30,35,37,36,38,24,31);(3,4,5,6)(7,9,12,21,14,22,13,11)(8,15,18,17,19,27,28,34)(10,20,25,35,26,36,29,24)(16,30,32,31,33,38,23,37);(4,6)(7,10)(8,16)(9,20)(11,24)(12,25)(13,29)(14,26)(15,30)(17,31)(18,32)(19,33)(21,35)(22,36)(23,28)(27,38)(34,37);(7,11,13,22,14,21,12,9)(8,17,28,15,19,34,18,27)(10,24,29,36,26,35,25,20)(16,31,23,30,33,37,32,38)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6372783);