Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4307937 |
| automorphism | Aut(G) order 2097152 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14762 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3215 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)(6,7)(10,13)(14,16)(18,22)(19,20);(5,6,8,7)(18,22)(23,24);(5,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22);(1,2)(3,4)(9,10)(11,14)(12,13)(15,16)(17,19,21,20)(18,23,22,24);(9,11,12,15)(10,14,13,16)(17,20,21,19)(18,24,22,23)Polycyclic presentation:
2^9|b^2=f,c^2=f,d^2=h,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4215268);