Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C29 |
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 | #36614 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C25×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
4547 conjugacy classes of subgroups; only the 454 normal subgroups are shown.
This group has 4547 conjugacy classes of subgroups — too many to draw. Only its 454 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) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,4)(5,6)(7,8)(10,13,14,15);(9,10,11,15,12,14,16,13);(2,4);(6,8)Polycyclic presentation:
2^9|a^2=h,b^2=eh,e^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,h^b=hi116 conjugacy classes — view the full character table.
| maximal subgroup of | #72286276 |
| maximal normal subgroup of | #72286276 |
| maximal quotient of | #72286337 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299815);