Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C8 | C23 |
| commutator | C23 | C22×C16 |
| socle, prime radical | C24 | C22×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
878 conjugacy classes of subgroups; only the 282 normal subgroups are shown.
This group has 878 conjugacy classes of subgroups — too many to draw. Only its 282 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,4)(5,6)(7,8)(17,18,19,22,20,23,25,28,21,24,26,29,27,30,31,32);(2,4)(9,10,12,14)(11,15,16,13);(6,8)(9,11,12,16)(10,13,14,15)Polycyclic presentation:
2^9|a^2=g,b^2=f,c^2=f,g^2=h,h^2=i,b^a=bd,c^a=ce,c^b=cf176 conjugacy classes — view the full character table.
| maximal subgroup of | #72287672, #72287692, #72287712, … |
| maximal normal subgroup of | #72287672, #72287692, #72287712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 56612);