Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C26×C4 |
| direct product | C2×#7942 |
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | #95 |
| commutator | C2×C4 | C22×C42 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C2×#95 |
| prime radical | C26 | D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
3562 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 18 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,17,18,15)(14,16);(4,8,10,9)(11,13)(12,15)(14,18)(16,17);(3,5,7,6)(4,9,10,8)128 conjugacy classes — view the full character table.
| maximal subgroup of | #72238876, C2×#110998, #72238881, … |
| maximal normal subgroup of | #72238876, C2×#110998, #72238881, … |
| maximal quotient of | C2×#94969, #72239120, C2×#127199, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 88311);