Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #233301 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×#121 |
| commutator, Frattini, cosocle | C2×C42 | C24 |
| prime radical | C2×#3253 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
1082 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6)(7,8)(9,11)(10,12)(14,19)(15,23)(16,17)(20,27)(21,22)(24,28);(1,2)(3,4)(6,11)(7,12)(8,9)(13,14)(15,24)(16,22)(17,19)(18,21)(20,26)(23,27)(25,28);(2,4)(5,7,9,12,10,11,8,6)(13,15,16,25,18,23,17,26)(14,20,19,27,21,28,22,24);(5,8,10,9)(6,7,12,11)(15,23)(20,28)(24,27)(25,26)Polycyclic presentation:
2^9|c^2=e,d^2=i,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fh,g^b=gh,d^c=dh,f^c=fhi62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 147617);