Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500596 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #3109 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| prime radical | #3150 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ #1726 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
541 conjugacy classes of subgroups; only the 144 normal subgroups are shown.
This group has 541 conjugacy classes of subgroups — too many to draw. Only its 144 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 — 4 generators on 36 points:
(6,13)(7,19,24,21)(8,10)(9,11)(14,30,32,34)(15,33)(17,18)(20,36,29,31)(22,28,23,35)(25,26);(1,2,3,4)(5,6,9,13,12,18,11,17)(7,20,23,32,24,29,22,14)(8,15,10,27,26,33,25,16)(19,30,28,31,21,34,35,36);(2,4)(5,7)(6,14)(8,21)(9,22)(10,28)(11,23)(12,24)(13,29)(15,31)(16,34)(17,20)(18,32)(19,26)(25,35)(27,30)(33,36);(5,8,9,10,12,26,11,25)(6,15,13,27,18,33,17,16)(7,21,22,28,24,19,23,35)(14,31,29,30,32,36,20,34)Polycyclic presentation:
2^9|a^2=eh,b^2=g,d^2=e,e^2=i,f^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,g^a=gi,h^a=hi,c^b=cg,f^b=fi,f^c=fhi,g^c=gi,h^c=hi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72239803, #72239811, #72239821, … |
| maximal normal subgroup of | #72239803, #72239811, #72239821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 413552);