Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #186146 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#379 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | #62812 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
3259 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,7,3,8)(9,10,14,18,15,11,13,17)(12,20,21,19,23,24,22,16);(2,6)(3,4)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,22)(21,24);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24);(2,7)(6,8)(9,13,15,14)(10,17,11,18)(12,21,23,22)(16,20,19,24)Polycyclic presentation:
2^9|a^2=g,d^2=h,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dhi,e^a=eh,e^b=ei,g^b=ghi,f^c=fh,g^c=gh74 conjugacy classes — view the full character table.
| maximal subgroup of | #72316711 |
| maximal normal subgroup of | #72316711 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 93501);