Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #480274 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×#99 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C22×D4 | C24 |
| prime radical | C2×#535 | C22 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C22⋊C4 ◁ C2×#518 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
1215 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(9,10)(11,19)(12,17)(13,15)(14,18,22,24)(16,20,23,21);(2,5)(6,8)(10,15)(11,12)(18,24)(20,21);(9,11,13,12)(10,16,15,23)(14,21,22,20)(17,24,19,18);(1,3,4,7)(2,6,5,8)(9,12,13,11)(10,17,15,19)(14,21,22,20)(16,24,23,18)Polycyclic presentation:
2^9|a^2=g,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,c^b=ci,d^b=di,f^c=fi,g^c=ghi,g^f=gi68 conjugacy classes — view the full character table.
| maximal subgroup of | #72286065, #72286124, #72286192 |
| maximal normal subgroup of | #72286065, #72286124, #72286192 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 401165);