Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2883 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C43 · Schur multiplier (H2) = C23×C43 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×C22⋊C4 |
| commutator | C22 | C2×C43 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2124 conjugacy classes of subgroups; only the 842 normal subgroups are shown.
This group has 2124 conjugacy classes of subgroups — too many to draw. Only its 842 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:
(1,2,4,3)(10,14)(12,19)(15,20)(18,23);(9,10,12,16,13,14,19,21)(11,15,18,22,17,20,23,24);(5,6,7,8)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(1,3,4,2)(11,17)(15,20)(18,23)(22,24)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=h,d^2=f,g^2=i,b^a=be,g^a=gi,e^b=ei,d^c=di164 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 260380);