Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6347429 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#120 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
3938 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,5,8)(3,7,4,6)(10,15)(12,13)(16,23)(17,18)(19,21)(22,24);(2,6)(3,4)(7,8)(9,10)(11,16)(12,15)(13,17)(14,18)(19,22)(20,24)(21,23);(9,11)(10,16)(12,19)(13,21)(14,20)(15,22)(17,23)(18,24);(9,12,14,13)(10,15,18,17)(11,19,20,21)(16,22,24,23);(1,3,5,4)(2,6,8,7)(9,12,14,13)(10,17,18,15)(11,19,20,21)(16,23,24,22)Polycyclic presentation:
2^9|a^2=hi,d^2=i,e^2=h,f^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,f^a=fh,e^b=eh,f^b=fh,g^b=gi,f^c=fi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72263904, #72263908, #72263924, … |
| maximal normal subgroup of | #72263904, #72263908, #72263924, … |
| maximal quotient of | #72264181 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6254650);