Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 8388608 |
| 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×#41 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
5170 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)(13,14,18,16)(15,20,23,19)(17,24,25,21)(22,26,28,27);(9,10,11,12)(13,15,18,23)(14,19,16,20)(17,22,25,28)(21,26,24,27);(6,7)(9,10,11,12)(13,16,18,14)(15,19,23,20)(17,24,25,21)(22,26,28,27);(1,2,3,4)(5,7,8,6)(9,10)(11,12)(13,15)(14,20)(16,19)(17,22)(18,23)(21,27)(24,26)(25,28);(13,17,18,25)(14,21,16,24)(15,22,23,28)(19,26,20,27)Polycyclic presentation:
2^9|a^2=h,b^2=f,c^2=f,d^2=i,e^2=h,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, 4837854);