Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #483987 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#442 |
| commutator | C2×C8 | C23×C4 |
| Frattini, cosocle | C22×C8 | C24 |
| prime radical | #33605 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ C22×#97 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
634 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(10,13)(11,19,21,24)(12,14)(16,22,23,20)(17,18);(1,3)(2,5)(4,8)(6,7)(9,10,14,17,15,18,12,13)(11,19,20,23,21,24,22,16);(3,8)(5,7)(9,11,13,19,14,22,10,16,15,21,17,24,12,20,18,23);(2,6)(5,7)(9,10,12,13,15,18,14,17)(11,16,20,19,21,23,22,24)Polycyclic presentation:
2^9|a^2=g,b^2=gh,c^2=f,d^2=hi,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=ce,d^b=di,f^b=fi,g^c=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72264735, #72264744, #72264749, … |
| maximal normal subgroup of | #72264735, #72264744, #72264749, … |
| maximal quotient of | #72309599, #72309724, #72309781, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 404229);