Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C2×C42 | C24 |
| socle | C23 | C2×C4×D4 |
| prime radical | C2×#121 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C2×#431 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,7,9,10,12,11)(14,18)(16,24)(19,25)(20,26)(21,28)(23,27);(13,14,16,18,17,20,24,26)(15,21,23,28,22,19,27,25);(2,4)(13,15)(14,19)(16,23)(17,22)(18,25)(20,21)(24,27)(26,28);(5,7,8,10,9,11,12,6)(15,22)(19,21)(23,27)(25,28)Polycyclic presentation:
2^9|a^2=g,b^2=e,d^2=g,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,c^b=ch,d^c=dh104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 118019);