Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4928752 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2682 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,3,4)(6,8)(11,15)(14,16)(18,19)(20,24);(10,13)(14,16)(17,18,21,19)(20,23,24,22);(5,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19,21,18)(20,22,24,23);(1,2)(3,4)(5,6)(7,8)(17,20,21,24)(18,22,19,23);(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^9|a^2=gi,b^2=f,c^2=g,d^2=f,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, 4836308);