Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #94871 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C24×C4 | C23 |
| socle | C23 | C2×#95 |
| prime radical | C26 | Q8 |
| LCS | G ▷ C22×C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C2 ▷ C1 | |
2994 conjugacy classes of subgroups; only the 530 normal subgroups are shown.
This group has 2994 conjugacy classes of subgroups — too many to draw. Only its 530 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10,12,16,14,17,13,22)(11,18,20,15,21,23,19,24);(5,6)(7,8)(9,11,13,20,14,21,12,19)(10,15,16,18,17,24,22,23);(2,4)(6,8)Polycyclic presentation:
2^9|a^2=g,b^2=h,d^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,h^a=hi,c^b=cf,g^b=gi104 conjugacy classes — view the full character table.
| Fitting subgroup of | #72267235 |
| maximal subgroup of | #72257798, #72258241, #72258255, … |
| maximal normal subgroup of | #72257798, #72258241, #72258255, … |
| maximal quotient of | #72267291 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 30705);