Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7304 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
| direct product | C2×#7304 |
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #342 |
| commutator | C24 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C24×D4 | C22 |
| LCS | G ▷ C24 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C25 ◁ C24×D4 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C23 ▷ C2 ▷ C1 | |
2480 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 18 points:
(1,2);(3,4)(5,9)(6,8)(7,10)(11,12)(13,17,18,15)(14,16);(3,5)(4,7,8,9)(6,10)(12,15)(16,17)68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219251, #72219253, #72219255, … |
| maximal normal subgroup of | #72219251, #72219253, #72219255, … |
| maximal quotient of | #72219251, #72226347, #72226348, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35227);