Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #242971 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#397 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#532 | C22 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
710 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(9,10,13,15,14,18,12,17)(11,19,21,24,20,23,22,16);(1,2)(3,5)(4,6)(7,8)(10,15)(11,20)(12,13)(16,23)(17,18)(19,24);(2,5)(3,8)(6,7)(9,11,12,21,14,20,13,22)(10,16,17,19,18,24,15,23);(2,6)(5,7)Polycyclic presentation:
2^9|a^2=f,c^2=e,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=ghi56 conjugacy classes — view the full character table.
| maximal subgroup of | #72238923, #72238925, #72238948 |
| maximal normal subgroup of | #72238923, #72238925, #72238948 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 150167);