Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94427 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1390 |
| commutator | C22×Q8 | C42 |
| Frattini, cosocle | #2297 | C22 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×Q8 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2297 ◁ G | |
| LECS | G ▷ #2297 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
325 conjugacy classes of subgroups; only the 42 normal subgroups are shown.
This group has 325 conjugacy classes of subgroups — too many to draw. Only its 42 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) = 32 |
A minimal faithful permutation representation — 2 generators on 24 points:
(2,5)(6,7)(9,10)(11,15,20,14)(12,18)(13,16)(17,21)(19,23,22,24);(1,2)(3,7,8,5)(4,6)(10,14,16,23,18,15,17,24)(11,19,20,22)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=i,e^2=fi,g^2=i,b^a=bc,c^a=cf,e^a=eg,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=dfi,g^b=gh,d^c=di,e^c=ehi,g^c=gi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72338940, #72338941, #72338944, … |
| maximal normal subgroup of | #72338940, #72338941, #72338944, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1690);