Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32932 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1230 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(6,8)(9,10,12,11)(13,14)(15,21)(16,25)(17,19)(18,23)(20,22)(24,27)(26,28);(2,4)(9,11,12,10)(13,14)(15,18)(16,20)(17,19)(21,23)(22,25)(24,26)(27,28);(13,15,22,28,17,24,16,23)(14,18,25,27,19,26,20,21);(5,6,7,8)(14,19)(18,26)(20,25)(21,27)Polycyclic presentation:
2^9|a^2=h,b^2=h,c^2=fi,d^2=gi,f^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 300938);