Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12506 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C8 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #493 |
| commutator | C4×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1970 |
| prime radical | #33713 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C82 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 122 normal subgroups are shown.
This group has 402 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) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,6)(7,8)(9,10,15,22,17,23,16,35)(11,19,26,39,28,36,27,40)(12,29,25,37,31,38,24,18)(13,32,30,21,14,34,33,20);(1,2,6,8,4,7,3,5)(9,11,13,24,15,26,30,12,17,28,14,25,16,27,33,31)(10,18,21,36,35,38,32,39,23,37,20,19,22,29,34,40);(9,12,30,27,16,24,13,28,17,31,33,26,15,25,14,11)(10,19,34,37,22,39,20,38,23,36,32,18,35,40,21,29)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=eh,d^2=gi,e^2=hi,f^2=i,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,h^a=hi,c^b=ci92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216249, #72216324, #72225340, … |
| maximal normal subgroup of | #72216249, #72216324, #72225340, … |
| maximal quotient of | #72216244, #72216251 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 48880);