Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13291 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1993 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| prime radical | C23 | #418 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ #1726 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
291 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 291 conjugacy classes of subgroups — too many to draw. Only its 102 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:
(1,2,4,7,5,3,6,8)(9,10,15,20,16,21,14,19)(11,22,24,32,25,31,18,13)(12,26,29,36,30,35,33,17)(23,28,34,38,39,40,27,37);(1,3,4,8,5,2,6,7)(9,11,13,21,14,24,31,19,16,25,32,10,15,18,22,20)(12,27,40,35,29,39,37,36,30,34,28,26,33,23,38,17);(9,12,16,30)(10,17)(11,23)(13,28,32,40)(14,29,15,33)(18,34)(19,26)(20,35)(21,36)(22,37,31,38)(24,27)(25,39)Polycyclic presentation:
2^9|a^2=f,b^2=def,c^2=eg,d^2=g,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=cg,e^b=ei,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72237291, #72239581, #72239582, … |
| maximal normal subgroup of | #72237291, #72239581, #72239582, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 49101);