Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3288 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C8 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | D42 |
| commutator | C2×C4 | C23×C8 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C22 | #2358 |
| prime radical | C2×#453 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,7,5,3,6,8)(9,10)(11,19)(12,20)(13,16)(14,23)(15,18)(17,24)(21,22);(1,3,4,8,5,2,6,7)(9,11,15,21)(10,13,18,12)(14,20,24,16)(17,22,23,19);(9,12)(10,11)(13,15)(14,22)(16,17)(18,21)(19,24)(20,23);(9,13)(10,16)(11,17)(12,15)(14,22)(18,20)(19,24)(21,23)Polycyclic presentation:
2^9|a^2=g,b^2=gh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=de,e^b=eh,f^c=fh,e^d=eh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 134131);