Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18746 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | C23×D4 | Q8 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1191 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8)(3,6,4,7)(9,10,13,17,14,16,12,21)(11,18,19,15,20,24,22,23);(2,6)(3,4)(7,8)(9,11,12,22,14,20,13,19)(10,15,21,18,16,23,17,24);(2,7)(3,4)(6,8);(10,16)(15,23)(17,21)(18,24)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dhi,e^a=eh,f^a=fi,e^b=eh,f^b=fi,e^c=ei,f^c=fi86 conjugacy classes — view the full character table.
| maximal subgroup of | #72285370, #72286392, #72286428 |
| maximal normal subgroup of | #72285370, #72286392, #72286428 |
| maximal quotient of | #72285673, #72319554, #72321487, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 100184);