Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18162 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#120 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#385 |
| prime radical | C23×D4 | Q8 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1084 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(9,10,13,16,14,17,12,21)(11,18,20,15,19,24,22,23);(1,2)(3,8)(4,6)(5,7)(9,11,13,20,14,19,12,22)(10,15,16,24,17,23,21,18);(3,7)(5,8);(2,6)(5,8)(11,19)(15,23)(18,24)(20,22)Polycyclic presentation:
2^9|a^2=g,b^2=g,e^2=h,g^2=hi,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=eh,f^b=fi,e^c=ei80 conjugacy classes — view the full character table.
| maximal subgroup of | #72265729, #72308407, #72308436 |
| maximal normal subgroup of | #72265729, #72308407, #72308436 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 99460);