Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13391 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2033 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C2×#405 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
650 conjugacy classes of subgroups.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2)(3,4)(6,12,15,14)(7,17,20,19)(8,22)(9,25)(10,24)(13,28)(16,23,27,26)(18,30)(21,34,31,32)(29,36);(5,6)(7,18)(8,23)(9,26)(10,27)(11,15)(12,25)(13,19)(14,22)(16,24)(17,29)(20,30)(21,35)(28,32)(31,33)(34,36);(2,4)(5,7,8,19,10,21,22,34,11,20,9,17,24,31,25,32)(6,13,12,18,16,28,26,33,15,29,14,30,27,36,23,35)Polycyclic presentation:
2^9|a^2=g,c^2=d,d^2=h,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,h^a=hi,c^b=cf,d^b=dhi,e^b=eh,f^b=fhi,g^b=gi,h^b=hi,f^c=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250041, #72250053, #72250104, … |
| maximal normal subgroup of | #72250041, #72250053, #72250104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55730);