Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25×C4 |
| direct product | C2×#11482 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
1050 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(3,4,6,9,7,10,5,8)(11,12)(13,21,24,36)(14,18)(15,28,30,19)(16,20)(17,23)(22,37,25,39)(26,41,42,34)(27,35)(29,32)(31,38)(33,40);(11,13,16,24)(12,17,20,32)(14,25,27,22)(15,29,30,23)(18,33,35,31)(19,36,28,21)(26,38,42,40)(34,37,41,39);(3,5,7,6)(4,8,10,9)(11,14,16,27)(12,18,20,35)(13,22,24,25)(15,26,30,42)(17,31,32,33)(19,34,28,41)(21,37,36,39)(23,38,29,40)92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 387304);