Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #360 |
| automorphism | ? |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | #102 |
| commutator | C8 | C82 |
| Frattini, cosocle | C42×C8 | C22 |
| socle | C22 | #1277 |
| prime radical | C23 | #341 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ C42×C8 ◁ G | |
| LECS | G ▷ C42×C8 ▷ C22×C4 ▷ C2 ▷ C1 | |
236 conjugacy classes of subgroups; only the 118 normal subgroups are shown.
This group has 236 conjugacy classes of subgroups — too many to draw. Only its 118 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 — 2 generators on 40 points:
(1,2)(3,5)(4,7)(6,8)(9,10,12,19,15,21,30,38,16,22,31,20,14,32,13,18)(11,23,25,36,28,34,40,39,29,37,24,17,27,33,26,35);(2,5,7,6)(9,11,13,26,14,27,31,24,16,29,30,40,15,28,12,25)(10,17,20,34,21,35,18,37,22,36,19,33,32,39,38,23)Polycyclic presentation:
2^9|a^2=d,b^2=e,c^2=fi,d^2=g,e^2=h,f^2=i,g^2=i,h^2=i,b^a=bc,c^a=cf,e^a=ef,f^a=fi,h^a=hi,c^b=ci128 conjugacy classes — view the full character table.
| maximal subgroup of | #72216225, #72216233, #72216234, … |
| maximal normal subgroup of | #72216225, #72216233, #72216234, … |
| maximal quotient of | #72227234 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 469);