Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12316 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
| direct product | C2×#12359 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator, Frattini, cosocle | C4×C8 | C24 |
| prime radical | C22×D8 | D4 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C42 ◁ C2×C8×Q8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
752 conjugacy classes of subgroups.
| minimal degree | μ(G) = 42 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 42 points:
(1,2);(4,8)(5,9)(6,10)(11,12,17,21)(13,22,25,41)(14,19,16,31)(15,27,30,39)(18,33,36,24)(20,26,38,29)(23,40,32,37)(28,34,42,35);(3,4,7,5)(6,9,10,8)(11,13,17,25)(12,18,21,36)(14,23,16,32)(15,28,30,42)(19,34,31,35)(20,37,38,40)(22,39,41,27)(24,26,33,29);(3,5,7,4)(6,8,10,9)(11,14,17,16)(12,19,21,31)(13,23,25,32)(15,26,30,29)(18,34,36,35)(20,27,38,39)(22,40,41,37)(24,28,33,42)80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 396223);