Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12307 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1616 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
390 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 390 conjugacy classes of subgroups — too many to draw. Only its 80 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,10,16,28,18,8,15,12,19,9,13,11,17,26,31)(7,20,23,33,29,36,21,14,25,32,22,34,24,30,27,35);(1,2,3,4)(6,13,19,16)(8,10)(9,26)(11,28)(14,30,35,36)(15,17,31,18)(20,34,32,33)(21,23)(22,27)(24,29);(5,7,12,25)(6,14,19,35)(8,23,26,22)(9,27,10,21)(11,29,28,24)(13,30,16,36)(15,20,31,32)(17,34,18,33)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=i,d^2=g,e^2=g,f^2=gi,g^2=i,h^2=i,b^a=bd,c^a=ce,e^a=eghi,h^a=hi,d^b=dg,e^b=ehi,f^b=fgi,g^b=gi,h^b=hi,d^c=dh,e^c=egi,f^c=fhi,g^c=gi,h^c=hi,e^d=ei,f^e=fi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72250959, #72250970, #72257006, … |
| maximal normal subgroup of | #72250959, #72250970, #72257006, … |
| maximal quotient of | #72265564 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44646);