Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1418 |
| automorphism | ? |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #357 |
| commutator | C16 | C4×C8 |
| Frattini, cosocle | C2×C4×C16 | C22 |
| socle | C22 | #1380 |
| prime radical | C23 | #358 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ C2×C4×C16 ◁ G | |
| LECS | G ▷ C2×C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
156 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 156 conjugacy classes of subgroups — too many to draw. Only its 82 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) = 256 |
A minimal faithful permutation representation — 2 generators on 72 points:
(1,2,4,7,5,8,3,6)(9,10,13,19,16,22,39,23)(11,24,26,59,29,49,30,68)(12,31,34,57,36,20,55,54)(14,40,37,52,35,18,48,51)(15,42,38,21,41,50,33,56)(17,43,44,25,47,61,53,32)(27,58,62,46,64,70,65,71)(28,66,60,69,63,72,67,45);(2,6,8,7)(9,11,14,27,41,63,55,43,39,26,48,62,38,67,36,25,16,29,35,64,15,28,34,61,13,30,37,65,33,60,12,32)(10,17,20,45,56,71,51,59,19,53,54,72,42,70,40,24,22,47,31,69,21,46,52,68,23,44,57,66,50,58,18,49)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fhi,d^2=g,e^2=fg,f^2=hi,h^2=i,b^a=bc,c^a=cf,e^a=efh,f^a=fh,h^a=hi,c^b=ci,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1246);