Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10374 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | #393 |
| commutator | C23×C4 | C22×C4 |
| Frattini, cosocle | C24×C4 | C23 |
| prime radical | C25 | C2×D4 |
| LCS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C24×C4 ◁ G | |
| LECS | G ▷ C24×C4 ▷ C22 ▷ C1 | |
792 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 28 points:
(6,7)(8,9)(11,12)(13,14)(15,21)(16,24,26,19)(17,20)(18,23)(22,27,25,28);(1,2)(3,4)(5,6,9,11,10,12,8,7)(13,15)(14,18,20,21)(16,25)(17,23)(19,28,24,27)(22,26);(2,4)(5,7,9,6,10,11,8,12)(13,16)(14,19)(15,22,23,25)(17,26)(18,27,21,28)(20,24)Polycyclic presentation:
2^9|a^2=g,b^2=e,c^2=dh,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,c^b=cf,d^b=di,f^c=fi,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 28059);