Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7943 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#95 |
| commutator | C2×C4 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C25 | M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C24×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
718 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,7,8,5)(4,6)(9,10,13,19,14,16,21,24,15,20,11,17,12,18,22,23);(1,3)(2,5)(4,8)(6,7)(10,16,20,18)(11,13)(17,24,19,23)(21,22);(9,11,14,22,15,13,12,21)(10,17,16,23,20,19,18,24)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=h,d^2=i,e^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fi,c^b=ci,e^b=egi104 conjugacy classes — view the full character table.
| maximal subgroup of | #72279532, #72279574, #72279601, … |
| maximal normal subgroup of | #72279532, #72279574, #72279601, … |
| maximal quotient of | C2×#126941 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34137);