Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #511896 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | C2×#575 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×M16 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
967 conjugacy classes of subgroups; only the 180 normal subgroups are shown.
This group has 967 conjugacy classes of subgroups — too many to draw. Only its 180 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 — 4 generators on 20 points:
(1,2)(3,4)(6,9)(7,15,17,12)(8,10)(13,14)(16,18,19,20);(5,6,8,9,11,13,10,14)(7,12,16,18,17,15,19,20);(5,7)(6,12)(8,16)(9,18)(10,19)(11,17)(13,15)(14,20);(2,4)(6,13)(9,14)(12,15)(18,20)Polycyclic presentation:
2^9|a^2=eh,b^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fh,h^a=hi,d^b=di,f^c=fhi,g^c=gi,h^c=hi,f^d=fi62 conjugacy classes — view the full character table.
| maximal subgroup of | #72219629, #72224588, #72250078, … |
| maximal normal subgroup of | #72219629, #72224588, #72250078, … |
| maximal quotient of | #72219622 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 419200);