Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #7426 |
| automorphism | ? |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #1351 |
| commutator | C2×C22⋊C4 | C42 |
| Frattini, cosocle | #2280 | C22 |
| socle | C2 | #7250 |
| prime radical | C25 | C22⋊C4 |
| LCS | G ▷ C2×C22⋊C4 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C23×C4 ◁ #2280 ◁ G | |
| LECS | G ▷ #2280 ▷ C24 ▷ C22 ▷ C2 ▷ C1 | |
406 conjugacy classes of subgroups; only the 60 normal subgroups are shown.
This group has 406 conjugacy classes of subgroups — too many to draw. Only its 60 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,5,9,6,10,19,26)(3,11,13,27,14,28,18,31)(4,8,16,23,17,24,30,32)(7,20,21,12,22,29,25,15);(1,3)(2,7)(4,15)(5,18)(6,14)(8,11,24,28)(9,25)(10,22)(12,17)(13,19)(16,29)(20,30)(21,26)(23,31,32,27)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=i,b^a=bc,c^a=cf,e^a=efg,h^a=hi,c^b=cg,d^b=di,f^b=fi,g^b=gh,e^c=eh,g^c=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72339725, #72339728, #72339729 |
| maximal normal subgroup of | #72339725, #72339728, #72339729 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 1493);