Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12077 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #442 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1580 |
| prime radical | C2×#135 | C2×C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×#346 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
297 conjugacy classes of subgroups; only the 104 normal subgroups are shown.
This group has 297 conjugacy classes of subgroups — too many to draw. Only its 104 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)(5,6,11,16,13,8,9,27,14,18,10,30,12,28,20,17)(7,19,24,15,25,32,23,21,26,36,22,34,31,35,29,33);(2,4)(5,7,20,23,12,31,10,24,14,26,9,29,13,25,11,22)(6,15,27,19,28,33,16,35,18,34,17,36,8,21,30,32);(5,8)(6,12)(7,21)(9,16)(10,27)(11,17)(13,18)(14,28)(15,31)(19,24)(20,30)(22,36)(23,32)(25,34)(26,33)(29,35)Polycyclic presentation:
2^9|a^2=f,b^2=fg,d^2=g,e^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,d^b=dg,e^b=egi,f^b=fi,g^b=gi,d^c=dgi,e^c=eg,f^c=fi,g^c=gi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72287284, #72319401, #72319447, … |
| maximal normal subgroup of | #72287284, #72319401, #72319447, … |
| maximal quotient of | #72287022 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 43109);