Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #117707 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #1439 |
| commutator | C2×C42 | C22×C4 |
| Frattini, cosocle | C22×C42 | C23 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
744 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(2,5,6,7)(9,10)(11,18)(12,22)(13,16)(14,17)(15,19)(20,23)(21,24);(9,11,12,19,14,20,13,21)(10,15,16,18,17,24,22,23);(1,2)(3,7)(4,6)(5,8)(11,19)(12,13)(15,23)(16,22)(18,24)(20,21)Polycyclic presentation:
2^9|a^2=g,b^2=d,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,e^b=eh,d^c=dh,e^c=ei,f^c=fh,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72239723, #72239729, #72239734, … |
| maximal normal subgroup of | #72239723, #72239729, #72239734, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 24903);