Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| 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 | C23 | C2×C4×C8 |
| Frattini, cosocle | C23×C8 | C23 |
| socle | C22 | C2×#336 |
| prime radical | C25 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ 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)(9,10,12,18,15,11,16,20,14,17,13,23,22,19,24,21);(1,3)(2,5)(4,8)(6,7)(9,11,13,20,15,17,24,23,14,19,12,21,22,10,16,18);(10,17)(11,19)(18,23)(20,21)Polycyclic presentation:
2^9|a^2=e,b^2=f,e^2=h,f^2=h,h^2=i,b^a=bd,c^a=ci,d^a=dg,f^a=fi,c^b=ci,d^b=di,e^b=eg104 conjugacy classes — view the full character table.
| maximal subgroup of | #72307594, #72307612, #72307616 |
| maximal normal subgroup of | #72307594, #72307612, #72307616 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34256);