Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #13316 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #447 |
| commutator | C22×C8 | C22×C4 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C22×#135 | C22 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
762 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,15)(11,19)(12,20)(13,22)(16,17)(18,23)(21,24);(10,16,17,15)(11,20)(12,19)(13,22)(18,21,24,23);(2,4)(9,10)(11,21)(12,23)(13,24)(14,17)(15,19)(16,20)(18,22)Polycyclic presentation:
2^9|a^2=g,b^2=d,e^2=hi,f^2=hi,h^2=i,b^a=bd,c^a=ce,e^a=eh,f^a=fh,h^a=hi,c^b=cf,e^b=ehi,f^b=fhi,h^b=hi,d^c=di,e^c=eh,f^c=fh,h^c=hi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72250249, #72250656, #72250666 |
| maximal normal subgroup of | #72250249, #72250656, #72250666 |
| maximal quotient of | #72307612 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 55316);