Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
| direct product | C2×#30495 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#3010 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
3132 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);(3,4)(5,6)(7,8,12,18,14,20,13,19)(9,21,23,34,25,33,24,15)(10,16,27,35,22,36,29,38)(11,28,26,17,30,37,31,32);(7,9,14,25)(8,15,20,34)(10,26,22,31)(11,29,30,27)(12,24,13,23)(16,28,36,37)(17,35,32,38)(18,33,19,21);(4,6)(7,10,14,22)(8,16,20,36)(9,11,25,30)(12,27,13,29)(15,32,34,17)(18,35,19,38)(21,28,33,37)(23,26,24,31);(4,6)(7,11)(8,17)(9,22)(10,25)(12,31)(13,26)(14,30)(15,16)(18,28)(19,37)(20,32)(21,38)(23,27)(24,29)(33,35)(34,36)68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7408102);