Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C42 · Schur multiplier (H2) = C214×C4 |
| direct product | C23×C4×C4⋊C4 |
| subgroup | G/H | |
|---|---|---|
| center | C25×C4 | C22 |
| commutator | C2 | C24×C42 |
| Frattini, cosocle | C23 | C26 |
| socle, prime radical | C26 | C23 |
| LCS | G ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
23602 conjugacy classes of subgroups.
| minimal degree | μ(G) = 18 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 6 generators on 18 points:
(1,2);(3,4);(5,6);(7,8,9,10);(12,14)(15,16,17,18);(11,12,13,14)320 conjugacy classes. The character table is too large to display.
| maximal subgroup of | C23×#1762, C23×C4×#121, C23×#2541, … |
| maximal normal subgroup of | C23×#1762, C23×C4×#121, C23×#2541, … |
| maximal quotient of | C23×C4×#95, C23×C4⋊#95, C23×#1773, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7532397);