Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #149789 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#354 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | #382 | C23 |
| prime radical | C24 | #99 |
| LCS | G ▷ C2×C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ #2285 ◁ G | |
| LECS | G ▷ #382 ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
306 conjugacy classes of subgroups; only the 62 normal subgroups are shown.
This group has 306 conjugacy classes of subgroups — too many to draw. Only its 62 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,3,6,4,5,7,8)(9,10)(11,21,23,34)(12,17)(13,28)(14,19,30,35)(15,20)(16,22,32,38)(18,29,36,27)(24,31,25,37)(26,40,39,33);(9,11,14,24,15,25,30,23)(10,16,19,32,20,33,35,40)(12,26,27,38,13,22,29,39)(17,34,18,21,28,31,36,37);(2,5)(6,8)(9,12,15,13)(10,17,20,28)(11,22,25,26)(14,27,30,29)(16,31,33,34)(18,35,36,19)(21,32,37,40)(23,38,24,39)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=i,d^2=h,e^2=g,f^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,e^b=efhi,e^d=ehi,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72292972, #72292988, #72293070, … |
| maximal normal subgroup of | #72292972, #72292988, #72293070, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 57000);