Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7418 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C23×C16 | C22 |
| socle | C22 | #1380 |
| prime radical | C24 | Q32 |
| LCS | G ▷ C2×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C23×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
223 conjugacy classes of subgroups; only the 72 normal subgroups are shown.
This group has 223 conjugacy classes of subgroups — too many to draw. Only its 72 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,18,15,21,34,20)(11,22,24,52,27,53,26,59)(12,28,31,51,33,61,66,60)(14,35,30,19,37,47,65,49)(16,38,39,55,41,54,50,23)(17,42,44,67,46,69,45,32)(25,57,56,70,58,71,36,43)(29,48,62,68,64,72,63,40);(2,5,7,6)(9,11,13,24,15,27,34,26)(10,16,18,39,21,41,20,50)(12,29,31,62,33,64,66,63)(14,36,30,25,37,56,65,58)(17,43,44,57,46,70,45,71)(19,48,47,68,49,72,35,40)(22,51,52,61,53,60,59,28)(23,42,38,67,55,69,54,32)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fh,d^2=i,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,g^a=gh,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi98 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1538);