Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14716 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(9,10)(11,15)(12,14)(13,16)(17,18,21,24)(19,22,20,23);(2,3)(5,6,7,8)(9,11,14,13)(10,15,12,16);(5,6)(7,8)(9,10,14,12)(11,16,13,15)(17,19,21,20)(18,22,24,23);(2,3)(9,12,14,10)(11,15,13,16)(17,18)(19,22)(20,23)(21,24);(1,3,4,2)(9,13,14,11)(10,15,12,16)(17,20,21,19)(18,23,24,22)Polycyclic presentation:
2^9|a^2=gi,b^2=h,c^2=fi,d^2=g,e^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2929090);