Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #7313 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1304 |
| commutator | C22×C4 | C4×C8 |
| Frattini, cosocle | C22×C4×D4 | C22 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ C22×C4×D4 ◁ G | |
| LECS | G ▷ C22×C4×D4 ▷ C24 ▷ C2 ▷ C1 | |
593 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 593 conjugacy classes of subgroups — too many to draw. Only its 70 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 — 2 generators on 24 points:
(1,2,3,5,4,6,7,8)(9,10)(11,17,19,24)(12,20,22,15)(13,16)(14,18)(21,23);(9,11)(10,14,16,23)(12,21,22,18)(13,19)(15,24)(17,20)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,d^2=h,b^a=bc,c^a=cf,e^a=egi,f^a=fi,c^b=cg,d^b=dfi,g^b=gi,d^c=di,e^c=ei56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 674);