Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #327 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C24×C42 |
| direct product | C2×#8048 |
| subgroup | G/H | |
|---|---|---|
| center | C2×C42 | C22⋊C4 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#95 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C42 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
2288 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4,6,9,7,10,5,8)(11,12,15,18,16,13,14,17);(4,8,10,9)(11,13,16,12)(14,18,15,17);(11,14,16,15)(12,17,13,18)152 conjugacy classes — view the full character table.
| maximal subgroup of | #72239189, #72239273, #72239292, … |
| maximal normal subgroup of | #72239189, #72239273, #72239292, … |
| maximal quotient of | #72239229, #72239504 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88810);