Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #133338 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1790 |
| commutator | C22×Q8 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | #46310 | C2 |
| LCS | G ▷ C22×Q8 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C23 ▷ C2 ▷ C1 | |
845 conjugacy classes of subgroups; only the 82 normal subgroups are shown.
This group has 845 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)(3,7,8,5)(4,6)(9,10)(11,21)(12,25,13,18)(14,19)(15,20)(16,22,33,27)(17,26,34,28)(23,37)(24,35)(29,31)(30,36)(32,38,40,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,22)(13,27)(14,29)(15,24)(17,31)(18,35)(19,32)(20,33)(21,25)(23,30)(26,39)(28,38)(34,37)(36,40);(9,12,15,13)(10,17,20,34)(11,22,24,27)(14,28,30,26)(16,31,33,37)(18,36,25,19)(21,32,35,40)(23,39,29,38)Polycyclic presentation:
2^9|a^2=f,c^2=i,d^2=i,e^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,h^a=hi,d^b=di,e^b=ehi,f^b=fgi,d^c=dh,e^c=ei,f^c=fi,e^d=ei,f^d=fi44 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 40536);