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) = C22×C4 |
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 | C24 | C2×M16 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,11,15,12,16,19,23,13,17,18,24,20,14,21,22);(2,5)(3,7)(10,14,17,16)(11,18)(15,23,24,22)(19,21);(9,11,12,19,13,18,20,21)(10,15,16,23,17,24,14,22)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=gh,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 | #72239316, #72239317, #72239329, … |
| maximal normal subgroup of | #72239316, #72239317, #72239329, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34161);