Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133922 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1782 |
| commutator | C25 | C22×C4 |
| Frattini, cosocle | C22×C22⋊C4 | C23 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C25 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C24 ◁ #2315 ◁ G | |
| LECS | G ▷ C22×C22⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
756 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 756 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) = 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)(11,19)(12,23)(13,26,28,17)(14,18)(15,21)(16,25,31,38)(20,34)(22,30,27,32)(24,37)(29,39,33,35)(36,40);(2,5)(3,7)(9,11)(10,15,18,19)(12,20)(13,27)(14,21)(16,29,31,33)(17,32,26,30)(22,28)(23,34,37,40)(24,36)(25,39)(35,38);(9,12,14,24)(10,16,18,31)(11,20,21,36)(13,25,28,38)(15,29,19,33)(17,23,26,37)(22,35,27,39)(30,40,32,34)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fg,d^c=di,f^c=fh,f^e=fi44 conjugacy classes — view the full character table.
| maximal subgroup of | #72284437, #72284468, #72284550, … |
| maximal normal subgroup of | #72284437, #72284468, #72284550, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 41370);