Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12087 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
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 | C23 | #444 |
| 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 | |
210 conjugacy classes of subgroups; only the 102 normal subgroups are shown.
This group has 210 conjugacy classes of subgroups — too many to draw. Only its 102 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)(5,6,10,16,11,18,29,35,12,19,9,25,26,17,27,15)(7,13,8,22,20,31,23,30,21,32,24,36,28,14,33,34);(2,4)(5,7,10,8,11,20,29,23,12,21,9,24,26,28,27,33)(6,13,16,22,18,31,35,30,19,32,25,36,17,14,15,34);(5,8,9,20,11,23,27,21,12,24,10,28,26,33,29,7)(6,14,15,22,17,32,25,34,19,31,35,36,18,13,16,30)Polycyclic presentation:
2^9|a^2=f,b^2=f,c^2=e,e^2=g,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eg,g^a=gi,c^b=cg,e^b=ei,f^c=fi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224944, #72224945, #72224951, … |
| maximal normal subgroup of | #72224944, #72224945, #72224951, … |
| maximal quotient of | C2×#140356, #72286882, #72286947, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47552);