Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C24 | C22×D4 |
| commutator | C22×C4 | C23×C4 |
| Frattini, cosocle | C23×C4 | C24 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ C2×C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1652 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)(9,10,12,11)(13,14)(15,25)(16,23)(17,20)(18,21)(19,24)(22,27)(26,28);(2,4)(5,6)(7,8)(9,11,12,10)(13,15,16,26,19,27,18,17)(14,20,21,22,24,28,23,25);(2,4)(6,8)(10,11)(13,16,19,18)(14,21,24,23)(15,26,27,17)(20,22,28,25);(2,4)(13,17,18,27,19,26,16,15)(14,22,23,20,24,25,21,28)Polycyclic presentation:
2^9|a^2=h,b^2=ef,c^2=i,d^2=efhi,e^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg116 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 285322);