Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #135928 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1864 |
| 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 | |
688 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)(6,13)(7,18,22,34)(8,11)(9,24)(10,23)(14,19,30,36)(15,32)(17,25)(20,21,26,33)(27,29,31,35);(5,6,8,13,12,17,11,25)(7,19,20,35,22,36,26,29)(9,16,10,15,23,28,24,32)(14,18,27,33,30,34,31,21);(2,4)(5,7)(6,14)(8,20)(9,21)(10,18)(11,26)(12,22)(13,27)(15,19)(16,29)(17,30)(23,33)(24,34)(25,31)(28,35)(32,36)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=i,e^2=h,f^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=eh,f^a=fhi,h^a=hi,c^b=cf,e^b=ehi,f^b=fh,h^b=hi,e^c=ehi,f^c=fh,g^c=gi,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250495, #72250508, #72250511, … |
| maximal normal subgroup of | #72250495, #72250508, #72250511, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55334);