Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#99 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C23×D4 | C23 |
| prime radical | C23×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×D4 ▷ C22 ▷ C2 ▷ C1 | |
1910 conjugacy classes of subgroups; only the 210 normal subgroups are shown.
This group has 1910 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(9,10)(11,15)(12,16,20,14)(13,17)(18,23)(19,22,24,21);(5,6,7,8)(10,14,17,16)(11,18)(15,22,23,21)(19,24);(2,4)(9,11,13,18)(10,15,17,23)(12,19,20,24)(14,21,16,22)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,h^a=hi,c^b=ci,f^b=fhi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72293046, #72293113, #72293115 |
| maximal normal subgroup of | #72293046, #72293113, #72293115 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53664);