Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11508 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#99 |
| commutator | C22×C4 | C22×C8 |
| Frattini, cosocle | C22×M16 | C23 |
| socle | C22 | #1459 |
| prime radical | C2×#532 | C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C22×#131 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
834 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 834 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) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(5,6,10,16,12,17,24,36,11,18,28,29,9,15,20,35)(7,19,22,32,21,31,8,14,23,33,26,34,27,13,25,30);(5,7)(6,13)(8,24)(9,27)(10,22)(11,23)(12,21)(14,16)(15,33)(17,19)(18,31)(20,25)(26,28)(29,30)(32,35)(34,36);(2,4)(5,8,12,26,11,25,9,22)(6,14,17,34,18,30,15,32)(7,20,21,10,23,24,27,28)(13,29,19,35,31,16,33,36)Polycyclic presentation:
2^9|a^2=f,c^2=h,d^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,c^b=ci,d^b=di,f^b=fgi,f^d=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72307676 |
| maximal normal subgroup of | #72307676 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36552);