Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4122169 |
| 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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2812 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(9,10)(11,13)(12,15)(14,16)(17,18,23,22)(19,20,24,21);(6,8)(10,11)(14,15)(17,19,23,24)(18,21,22,20);(5,6)(7,8)(12,16)(14,15)(17,20,23,21)(18,19,22,24);(1,2)(3,4)(9,11,13,10)(12,14,16,15)(17,21,23,20)(18,24,22,19);(9,12,13,16)(10,14,11,15)(17,22,23,18)(19,21,24,20)Polycyclic presentation:
2^9|a^2=g,b^2=g,c^2=g,d^2=f,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=df,e^b=eg,e^c=ef,e^d=ef92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4029544);