Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32706 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C22×C22⋊C4 |
| prime radical | #36624 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C23×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1382 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,15)(13,16)(17,18,21,24,22,20,19,23);(1,3)(2,5)(4,8)(6,7)(9,11,13,12)(10,14,16,15)(17,19,22,21)(18,23,20,24);(3,8)(5,7)(10,15)(11,12)(14,16)(17,20)(18,22)(19,24)(21,23);(2,6)(5,7)(18,20)(23,24)Polycyclic presentation:
2^9|a^2=g,b^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cehi,d^c=di,f^c=fh80 conjugacy classes — view the full character table.
| maximal subgroup of | #72239119 |
| maximal normal subgroup of | #72239119 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 121415);