Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32929 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #36644 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×C43 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2538 conjugacy classes of subgroups; only the 490 normal subgroups are shown.
This group has 2538 conjugacy classes of subgroups — too many to draw. Only its 490 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,4)(6,8)(9,10,12,15,13,16,11,14);(10,14,16,15);(1,2,3,4);(5,6,7,8)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=f,d^2=g,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei152 conjugacy classes — view the full character table.
| maximal subgroup of | #72225926, #72225927, #72225944, … |
| maximal normal subgroup of | #72225927, #72225944, #72226040, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300971);