Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1714 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #127 |
| commutator | C42 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1954 |
| prime radical | #3147 | C4 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
430 conjugacy classes of subgroups; only the 134 normal subgroups are shown.
This group has 430 conjugacy classes of subgroups — too many to draw. Only its 134 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) = 128 |
A minimal faithful permutation representation — 3 generators on 40 points:
(2,5)(3,6)(7,8)(9,10,15,21,16,20,31,37,13,19,23,18,14,32,12,29)(11,22,27,36,28,38,24,39,25,17,33,34,26,35,30,40);(1,2,6,8,4,7,3,5)(9,11,23,33,16,28,12,30,13,25,15,27,14,26,31,24)(10,17,18,36,20,35,29,39,19,22,21,34,32,38,37,40);(9,12,14,23,13,31,16,15)(10,18,20,29,19,21,32,37)(11,24,26,27,25,30,28,33)(17,34,35,40,22,36,38,39)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,c^2=e,d^2=g,e^2=i,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,c^b=ci104 conjugacy classes — view the full character table.
| maximal subgroup of | #72225174, #72225220, #72225442, … |
| maximal normal subgroup of | #72225174, #72225220, #72225442, … |
| maximal quotient of | #72225171 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 47712);