Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13360 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2021 |
| commutator | C2×#41 | C22×C4 |
| Frattini, cosocle | C22×#41 | C23 |
| prime radical | C2×#585 | C22 |
| LCS | G ▷ C2×#41 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×#41 ▷ C22 ▷ C2 ▷ C1 | |
1355 conjugacy classes of subgroups; only the 130 normal subgroups are shown.
This group has 1355 conjugacy classes of subgroups — too many to draw. Only its 130 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,3,4)(6,12)(7,16,19,21)(8,10)(13,28,30,36)(14,27)(15,23)(17,31,26,29)(18,35,24,34)(20,25)(32,33);(5,6,10,23,11,15,8,12)(7,17,24,30,19,26,18,13)(9,14,25,32,22,33,20,27)(16,28,35,29,21,36,34,31);(5,7)(6,13)(8,18)(9,21)(10,24)(11,19)(12,26)(14,31)(15,30)(16,22)(17,23)(20,35)(25,34)(27,36)(28,32)(29,33)Polycyclic presentation:
2^9|a^2=g,b^2=di,d^2=i,e^2=i,f^2=i,b^a=bd,c^a=ce,d^a=di,f^a=fhi,c^b=cf,e^b=eh,f^b=fi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^e=fi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72311939, #72311953, #72311964, … |
| maximal normal subgroup of | #72311939, #72311953, #72311964, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55378);