Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| 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 | |
1526 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(9,10,12,11)(13,14,17,20)(15,22,24,19)(16,26,21,18)(23,25,27,28);(1,2,4,3)(5,6)(7,8)(9,11,12,10)(13,15,17,24)(14,19,20,22)(16,23,21,27)(18,25,26,28);(6,8)(13,16,18,14,17,21,26,20)(15,23,25,19,24,27,28,22);(1,3,4,2)(14,20)(15,24)(16,21)(25,28)Polycyclic presentation:
2^9|a^2=h,b^2=ehi,c^2=f,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=cg,d^b=di,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 272764);