Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
| direct product | C2×#12869 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D8 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| socle | C23 | C2×#102 |
| prime radical | C25 | SD16 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C24×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1098 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)(5,8)(6,7)(9,10)(11,12,15,20,17,21,16,31)(13,22,24,38,25,39,32,36)(14,26,28,37,30,41,29,19)(18,27,33,40,35,42,34,23);(4,7)(8,10);(3,5,6,9)(4,8,7,10)(11,13)(12,18)(14,27)(15,24)(16,32)(17,25)(19,36)(20,33)(21,35)(22,26)(23,29)(28,40)(30,42)(31,34)(37,38)(39,41)116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401100);