Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8760 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #381 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | C2×#381 |
| prime radical | C24 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
965 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,5,8,6,3,4,7)(9,10)(11,16,12,23)(13,18)(14,20)(15,19)(17,21,24,22);(9,11,14,12)(10,16,20,23)(13,21,15,22)(17,18,24,19);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,17)(13,22,15,21)(16,18)(19,23)(20,24);(1,4,6,5)(2,7,3,8)(9,13,14,15)(10,18,20,19)(11,21,12,22)(16,24,23,17)Polycyclic presentation:
2^9|a^2=g,b^2=h,c^2=e,d^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=gh,f^c=fi,g^c=gh86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 97510);