Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #243068 |
| 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 | C23×D4 | D4 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6,5,7,3,8)(9,10,13,15,14,17,12,18)(11,19,21,24,20,23,22,16);(10,15)(11,20)(12,13)(16,23)(17,18)(19,24);(1,2)(3,6)(4,8)(5,7)(9,11,12,21,14,20,13,22)(10,16,18,19,17,24,15,23);(1,3,5,4)(2,6,7,8)(10,17)(11,20)(15,18)(21,22)Polycyclic presentation:
2^9|a^2=f,c^2=e,d^2=hi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=di,g^a=gh,c^b=cg,e^b=eh,f^b=fh,d^c=dh,f^c=fi,g^c=gh56 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 150270);