Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
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 | |
1756 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(9,10,11,12)(13,14)(15,23)(16,21)(17,19)(18,22)(20,24)(25,28)(26,27);(1,2,4,3)(5,6)(7,8)(14,19)(16,17)(20,23)(21,22)(24,26)(27,28);(6,8)(13,15,18,25)(14,20,22,27)(16,24,17,26)(19,23,21,28);(1,3,4,2)(13,16)(15,24)(17,18)(19,21)(23,28)(25,26)Polycyclic presentation:
2^9|a^2=h,b^2=efi,c^2=i,d^2=efi,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,f^c=fi,e^d=ei,f^d=fi92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 268703);