Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | ? |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C23×C4 | C24 |
| commutator, socle | C24 | C23×C4 |
| prime radical | C23×D4 | C2×C4 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
1666 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)(5,6)(7,8)(13,14,18,22,17,21,16,20)(15,19,23,26,24,27,25,28);(1,2,4,3)(10,12)(13,14,16,20,17,21,18,22)(15,19,25,28,24,27,23,26);(6,8)(9,10,11,12)(13,15)(14,19)(16,25)(17,24)(18,23)(20,28)(21,27)(22,26);(1,3,4,2)(13,16,17,18)(14,20,21,22)(15,23,24,25)(19,26,27,28)Polycyclic presentation:
2^9|a^2=h,b^2=eh,c^2=g,d^2=e,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,d^c=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 275511);