Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
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 | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1702 conjugacy classes of subgroups; only the 506 normal subgroups are shown.
This group has 1702 conjugacy classes of subgroups — too many to draw. Only its 506 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 — 4 generators on 24 points:
(2,4)(6,8)(9,10,13,17,14,12,21,16)(11,15,20,23,18,19,24,22);(1,2,3,4)(9,11)(10,15)(12,19)(13,20)(14,18)(16,22)(17,23)(21,24);(5,6,7,8)(11,18)(15,19)(20,24)(22,23);(9,12)(10,14)(11,19)(13,16)(15,18)(17,21)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,g^a=gi,c^b=ci,g^d=gi104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299648);