Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #129558 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | #3054 | C4 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
471 conjugacy classes of subgroups; only the 94 normal subgroups are shown.
This group has 471 conjugacy classes of subgroups — too many to draw. Only its 94 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,7,11,9,14,17,20,10,12,15,19,8,18,16,13);(6,11,14,20,12,19,18,13);(2,4)(6,12)(7,15)(14,18)(16,17)Polycyclic presentation:
2^9|a^2=f,b^2=dfg,d^2=gi,f^2=h,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,c^b=cgh,d^c=di,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72214871, #72219588, #72219589 |
| maximal normal subgroup of | #72214871, #72219588, #72219589 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44862);