Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 25165824 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C212 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×2−1+4 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
17322 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)(13,14,17,18)(15,21,22,19)(16,24,25,20)(23,26,28,27);(6,7)(14,18)(15,22)(20,24)(23,28);(10,11)(13,14)(15,21)(16,20)(17,18)(19,22)(23,27)(24,25)(26,28);(1,2,3,4)(5,7,8,6)(9,11,12,10)(13,15,17,22)(14,19,18,21)(16,23,25,28)(20,26,24,27);(13,16,17,25)(14,20,18,24)(15,23,22,28)(19,26,21,27)Polycyclic presentation:
2^9|a^2=h,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, 4837851);