Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2×C4 |
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 | |
245 conjugacy classes of subgroups; only the 90 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 90 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)(5,6,9,12,10,17,27,15,11,18,8,24,25,16,26,14)(7,19,21,13,20,30,34,32,23,29,22,33,36,28,35,31);(6,12,16,14,18,24,17,15)(7,20,23,36)(8,9)(13,29,32,28,33,19,31,30)(21,35,22,34)(26,27);(2,4)(5,7,8,21,10,20,26,34,11,23,9,22,25,36,27,35)(6,13,14,29,16,31,24,30,18,33,15,19,17,32,12,28)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=d,d^2=g,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eg,g^a=gi,c^b=cg,d^b=di,e^b=ei,f^b=fi,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224944, #72224945, #72224958, … |
| maximal normal subgroup of | #72224944, #72224945, #72224958, … |
| maximal quotient of | C2×#140191, #72286947, #72286997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47387);