Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30484 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| socle | C22 | #2358 |
| prime radical | C22×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
934 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(10,16)(11,18,22,23)(12,20,17,24)(19,21);(1,2,4,5)(3,8,7,6)(9,10,13,16)(11,20,22,24)(12,23,17,18)(14,19,15,21);(9,11,14,23,13,22,15,18)(10,17,19,24,16,12,21,20);(1,3,4,7)(2,6,5,8)(9,12,13,17)(10,18,16,23)(11,21,22,19)(14,20,15,24)Polycyclic presentation:
2^9|a^2=h,b^2=e,c^2=fh,d^2=e,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gi,f^c=fi,g^c=gi,h^c=hi,f^d=fi,h^d=hi80 conjugacy classes — view the full character table.
| maximal quotient of | #72210005, #72238811, #72238812, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 296435);