Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12298 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1613 |
| commutator | C2×M16 | C22×C4 |
| Frattini, cosocle | C22×M16 | C23 |
| prime radical | C24 | C2×SD16 |
| LCS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
388 conjugacy classes of subgroups; only the 80 normal subgroups are shown.
This group has 388 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) = 128 |
A minimal faithful permutation representation — 3 generators on 36 points:
(2,4)(5,6,9,17,10,18,28,15,12,19,27,13,11,30,8,16)(7,20,22,33,23,34,29,36,25,35,26,14,24,31,21,32);(1,2,3,4)(6,13)(8,9)(10,11)(14,20)(15,18)(16,30)(17,19)(21,22)(23,24)(26,29)(27,28)(31,32)(33,35)(34,36);(5,7,12,25)(6,14,19,33)(8,26,28,22)(9,29,27,21)(10,24,11,23)(13,20,17,35)(15,34,16,31)(18,36,30,32)Polycyclic presentation:
2^9|a^2=f,b^2=dfgh,c^2=i,d^2=gi,e^2=g,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=di,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 | #72250955, #72250970, #72257005, … |
| maximal normal subgroup of | #72250955, #72250970, #72257005, … |
| maximal quotient of | #72265564 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 44651);