Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #94343 |
| 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 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C23×C8 ◁ C23×C16 ◁ G | |
| LECS | G ▷ C23×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
220 conjugacy classes of subgroups; only the 70 normal subgroups are shown.
This group has 220 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) = 72 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,20,17,23,16,44)(11,24,26,59,30,62,27,60)(12,31,34,69,38,54,37,58)(14,40,33,22,15,42,39,21)(18,45,46,25,50,64,47,66)(19,51,53,36,55,63,70,35)(28,67,43,52,29,68,41,61)(32,56,71,49,65,57,72,48);(2,5,7,6)(9,11,13,27,17,30,16,26)(10,18,20,47,23,50,44,46)(12,32,34,72,38,65,37,71)(14,41,33,29,15,43,39,28)(19,52,53,67,55,61,70,68)(21,56,40,48,22,57,42,49)(24,58,60,31,62,69,59,54)(25,63,45,35,66,51,64,36)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=i,e^2=i,f^2=hi,g^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efghi,f^a=fh,g^a=gh,h^a=hi,c^b=cg,d^b=di,f^b=fh,g^b=gh,h^b=hi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1542);