Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11669 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #418 |
| commutator | C2×C8 | C22×C8 |
| Frattini, cosocle | C2×C4×C8 | C23 |
| socle | C22 | #1490 |
| prime radical | C2×#575 | C4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C8 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
829 conjugacy classes of subgroups; only the 140 normal subgroups are shown.
This group has 829 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,10,14,12,15,20,13,9,16,8,19,11,7,17,18);(5,7)(6,12)(8,13)(9,15)(10,18)(11,16)(14,20)(17,19);(2,4)(8,10)(13,18)(14,19)(17,20)Polycyclic presentation:
2^9|a^2=f,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,d^b=dg,e^b=ei,g^b=gi,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72219605, #72219629, #72219637 |
| maximal normal subgroup of | #72219605, #72219629, #72219637 |
| maximal quotient of | #72219595, #72219596 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 36707);