Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #105580 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C4×C22⋊C4 |
| commutator | C22×C4 | C2×C42 |
| Frattini, cosocle | C22×C42 | C23 |
| socle | C22 | C22⋊#95 |
| prime radical | C23×D4 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C42 ▷ C22 ▷ C1 | |
1294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(9,10,13,17,14,18,12,15)(11,19,20,24,21,16,22,23);(2,5)(3,7)(10,15,18,17)(16,23,19,24);(2,5)(3,7)(9,11)(10,16)(12,20)(13,22)(14,21)(15,23)(17,24)(18,19)Polycyclic presentation:
2^9|a^2=f,b^2=g,d^2=hi,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,f^b=fi,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72233843, #72233844, #72233845, … |
| maximal normal subgroup of | #72233843, #72233844, #72233845, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 12778);