Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #115163 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1438 |
| commutator, prime radical | C25 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
1766 conjugacy classes of subgroups; only the 158 normal subgroups are shown.
This group has 1766 conjugacy classes of subgroups — too many to draw. Only its 158 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)(3,5)(4,6)(7,8)(9,10,13,18,14,17,21,24)(11,16,19,23,12,15,20,22);(5,8)(9,11,14,12)(10,15,17,16)(13,20,21,19)(18,23,24,22);(1,3)(2,5)(4,7)(6,8)(9,12,14,11)(10,16,17,15)(13,19,21,20)(18,22,24,23)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,g^b=gh,d^c=di,g^c=gh62 conjugacy classes — view the full character table.
| maximal subgroup of | #72303886, #72303887, #72303897 |
| maximal normal subgroup of | #72303886, #72303887, #72303897 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 22364);