Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19279 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×#121 |
| commutator | C42 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#388 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
838 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,7)(3,8)(4,5)(9,10,13,15,14,18,12,17)(11,19,22,24,20,16,21,23);(1,2,4,8,6,3,5,7)(10,15,18,17)(11,20)(16,24,19,23)(21,22);(1,3,5,8,6,2,4,7)(9,11,13,22,14,20,12,21)(10,16,17,24,18,19,15,23);(1,4,6,5)(2,8,3,7)(9,12,14,13)(10,17,18,15)(11,21,20,22)(16,24,19,23)Polycyclic presentation:
2^9|a^2=g,b^2=egh,c^2=e,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,c^b=ch,f^c=fhi80 conjugacy classes — view the full character table.
| maximal subgroup of | #72285487, #72286392, #72286469 |
| maximal normal subgroup of | #72285487, #72286392, #72286469 |
| maximal quotient of | #72285942, #72286002, #72321504 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 110332);