Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C210 |
| direct product | C22×#1490 |
| subgroup | G/H | |
|---|---|---|
| center | C23×C4 | C2×D4 |
| commutator | C2×C4 | C24×C4 |
| Frattini, cosocle | C42 | C25 |
| socle | C24 | C2×C22⋊C4 |
| prime radical | C24×D4 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
6394 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2);(3,4);(5,6)(7,11)(8,10)(9,12)(13,14,15,18,16,17,19,20);(6,10)(7,8)(11,12);(5,7,9,8)(6,11,12,10)(14,17)(18,20)152 conjugacy classes — view the full character table.
| maximal subgroup of | C22×#18524, C22×#18529, C22×#18525, … |
| maximal normal subgroup of | C22×#18524, C22×#18529, C22×#18525, … |
| maximal quotient of | #72219678, C2×#184863, #72309359, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6250111);