Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#388 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
810 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(9,10,13,17,14,12,21,16)(11,15,19,23,20,18,24,22);(1,2,5,7,6,8,4,3)(9,11,13,19,14,20,21,24)(10,15,17,23,12,18,16,22);(1,3,4,8,6,7,5,2)(9,12)(10,14)(11,18)(13,16)(15,20)(17,21)(19,22)(23,24);(1,4,6,5)(2,7,8,3)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dh,d^c=dh,f^c=fhi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72342639, #72342644, #72342649 |
| maximal normal subgroup of | #72342639, #72342644, #72342649 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 110326);