Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D4×#392 |
| automorphism | Aut(G) order 4194304 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C29 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#121 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3194 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)(11,15)(12,16)(13,14)(17,18,22,24)(19,23,20,21);(6,8)(10,14)(12,15)(17,19,22,20)(18,23,24,21);(5,6)(7,8)(18,24)(21,23);(1,2)(3,4)(9,11,13,16)(10,15,14,12)(17,20,22,19)(18,21,24,23);(9,12,13,15)(10,16,14,11)(17,21)(18,19)(20,24)(22,23)Polycyclic presentation:
2^9|a^2=g,b^2=g,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg110 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4213642);