Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2180 |
| automorphism | ? |
| homology | Abelianization (H1) = C43 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #442 |
| commutator | C8 | C43 |
| Frattini, cosocle | C23×C8 | C23 |
| prime radical | C24 | AGL(1,ℤ/8) |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C4 ▷ C2 ▷ C1 | |
682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,4)(5,6,7,8)(10,15)(11,20)(12,21)(13,22)(16,23)(17,19)(18,24);(2,4)(9,10,12,17,13,19,11,15,14,18,20,23,22,16,21,24);(10,16,18,19)(11,21)(12,20)(15,23,24,17)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=g,d^2=hi,f^2=h,h^2=i,b^a=bd,d^a=dh,f^a=fhi,h^a=hi,c^b=ch,g^b=gi,d^c=di,f^c=fi92 conjugacy classes — view the full character table.
| maximal subgroup of | #72216446, #72216458, C4×#12647, … |
| maximal normal subgroup of | #72216446, #72216458, C4×#12647, … |
| maximal quotient of | #72240743, #72280067, C2×#146344, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 53540);