Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #198126 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#386 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3115 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1236 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,7,6,3,4,8)(9,10,12,16,13,15,17,23)(11,14,19,22,18,21,20,24);(1,3)(2,6)(4,8)(5,7)(9,11,17,19,13,18,12,20)(10,14,16,24,15,21,23,22);(1,2)(3,6)(4,7)(5,8);(1,4,6,5)(2,7,3,8)(10,15)(14,21)(16,23)(22,24)Polycyclic presentation:
2^9|a^2=g,b^2=egi,d^2=hi,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fhi,g^c=ghi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 105493);