Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12711 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1781 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C2×#117 | C23 |
| prime radical | C22×#41 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#520 ◁ G | |
| LECS | G ▷ C2×#117 ▷ C23 ▷ C2 ▷ C1 | |
873 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 873 conjugacy classes of subgroups — too many to draw. Only its 122 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(1,2,4,7,5,8,3,6)(9,10)(11,21)(12,25,13,18)(14,19)(15,20)(16,22,33,27)(17,26,35,28)(23,31)(24,36)(29,40)(30,34)(32,38,37,39);(9,11,15,24)(10,16,20,33)(12,22,13,27)(14,29,30,23)(17,31,35,40)(18,36,25,21)(19,37,34,32)(26,39,28,38);(1,3,5,4)(2,6,8,7)(9,12)(10,17)(11,22)(13,15)(14,28)(16,31)(18,19)(20,35)(21,32)(23,39)(24,27)(25,34)(26,30)(29,38)(33,40)(36,37)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=di,e^b=ei,f^b=fgi,d^c=di,e^c=ei,f^c=fi,f^d=fi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36545);