Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C211 |
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×#585 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×#41 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
16110 conjugacy classes of subgroups; only the 698 normal subgroups are shown.
This group has 16110 conjugacy classes of subgroups — too many to draw. Only its 698 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:
(1,2)(3,4)(5,6)(7,8)(9,10,13,17,14,11,18,16)(12,15,21,23,20,19,24,22);(9,11)(10,14)(12,19)(13,16)(15,20)(17,18)(21,22)(23,24);(2,4)(9,12)(10,15)(11,19)(13,21)(14,20)(16,22)(17,23)(18,24);(6,8)(12,20)(15,19)(21,24)(22,23)Polycyclic presentation:
2^9|a^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,e^b=ei,d^c=di104 conjugacy classes — view the full character table.
| maximal subgroup of | #72286571 |
| maximal normal subgroup of | #72286571 |
| maximal quotient of | #72286783 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 299320);