Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #352352 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #2856 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | #19292 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
729 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(10,16)(11,18)(12,20)(13,14)(17,24)(22,23);(2,6)(5,8)(9,10,13,16,15,18,14,11)(12,17,20,21,23,24,22,19);(1,2,3,5,4,6,7,8)(9,11,13,10,15,16,14,18)(12,21,20,24,23,19,22,17);(2,6)(5,8)(9,12)(10,17)(11,19)(13,20)(14,22)(15,23)(16,21)(18,24)Polycyclic presentation:
2^9|b^2=e,c^2=fh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ci,d^c=di,g^d=gh62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 259552);