Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #213728 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2308 |
| commutator | C24 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C22×2−1+4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
2404 conjugacy classes of subgroups; only the 346 normal subgroups are shown.
This group has 2404 conjugacy classes of subgroups — too many to draw. Only its 346 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 — 4 generators on 36 points:
(1,2)(3,4)(5,6,11,16,10,15,20,17)(7,12,19,8,13,23,26,22)(9,14,25,31,24,30,27,32)(18,28,33,35,29,36,34,21);(2,4)(5,7,10,13)(6,12,15,23)(8,16,22,17)(9,18,24,29)(11,26,20,19)(14,28,30,36)(21,32,35,31)(25,34,27,33);(5,8,20,23,10,22,11,12)(6,13,16,19,15,7,17,26)(9,21,25,36,24,35,27,28)(14,29,32,34,30,18,31,33);(2,4)(5,9)(6,14)(7,18)(8,21)(10,24)(11,27)(12,28)(13,29)(15,30)(16,32)(17,31)(19,34)(20,25)(22,35)(23,36)(26,33)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=fhi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,g^a=gi,c^b=cg,h^b=hi,e^c=ei,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 281208);