Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11692 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
| direct product | C2×#11740 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C24 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
926 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);(4,8,9,10)(11,12,15,20,16,19,28,35)(13,21,23,36,24,17,29,31)(14,18,26,34,27,33,39,42)(22,32,37,41,25,30,38,40);(3,4)(5,8)(6,10)(7,9)(11,13,16,24)(12,17,19,21)(14,25,27,22)(15,23,28,29)(18,32,33,30)(20,36,35,31)(26,38,39,37)(34,40,42,41);(3,5,7,6)(4,8,9,10)(11,14,16,27)(12,18,19,33)(13,22,24,25)(15,26,28,39)(17,30,21,32)(20,34,35,42)(23,37,29,38)(31,40,36,41)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 388733);