Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94202 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1380 |
| commutator | C2×C16 | C42 |
| Frattini, cosocle | C23×C16 | C22 |
| prime radical | C24 | Q32 |
| LCS | G ▷ C2×C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C23×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
178 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 178 conjugacy classes of subgroups — too many to draw. Only its 60 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)(3,7,8,5)(4,6)(9,10,13,20,17,23,16,41)(11,24,26,54,30,55,29,65)(12,21,32,49,34,51,33,66)(14,19,35,47,38,48,37,50)(15,39,36,52,40,67,31,22)(18,42,43,58,46,57,68,25)(27,53,59,70,62,71,61,45)(28,63,60,69,64,72,56,44);(1,3)(2,5)(4,8)(6,7)(9,11,14,27,15,28,32,42,13,26,35,59,36,60,34,58,17,30,38,62,40,64,33,57,16,29,37,61,31,56,12,25)(10,18,21,44,22,45,50,65,41,68,66,72,67,71,48,55,23,46,51,69,52,70,47,54,20,43,49,63,39,53,19,24)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fgh,d^2=i,e^2=fg,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efgh,f^a=fh,h^a=hi,d^b=dg92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1409);