Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19072 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
998 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,6)(3,4)(7,8)(9,10,14,18,15,19,13,21)(11,20,17,16,23,24,22,12);(1,2)(3,6)(4,8)(5,7)(9,11,14,17,15,23,13,22)(10,16,18,24,19,12,21,20);(3,4)(6,8)(9,12)(10,17)(11,21)(13,24)(14,20)(15,16)(18,23)(19,22);(1,3,5,4)(2,6,7,8)(9,13,15,14)(10,18,19,21)(11,22,23,17)(12,20,16,24)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=h,e^2=h,f^2=hi,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fhi,e^b=eh,f^b=fi,d^c=dh,e^c=ei,f^c=fhi80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100358);