Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 12582912 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C210 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5077 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 28 points:
(2,4)(5,6)(7,8)(9,10)(11,12)(15,21)(16,22)(17,25)(19,26);(6,7)(10,11)(13,14,18,20)(15,22,21,16)(17,19,25,26)(23,28,27,24);(10,11)(13,15,18,21)(14,16,20,22)(17,23,25,27)(19,24,26,28);(1,2)(3,4)(5,7,8,6)(9,11,12,10)(13,16)(14,15)(17,24)(18,22)(19,23)(20,21)(25,28)(26,27);(13,17)(14,19)(15,23)(16,24)(18,25)(20,26)(21,27)(22,28)Polycyclic presentation:
2^9|b^2=h,c^2=h,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,d^c=dg104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4836042);