Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3069037 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #15320 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2281 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,6)(3,5,7,8)(10,11)(12,16)(17,18,21,22)(19,23,24,20)(25,26)(27,31)(28,30)(29,32);(9,10,13,11)(12,15,16,14)(18,22)(20,23)(25,27,30,29)(26,31,28,32);(10,11)(14,15)(17,19,21,24)(18,23,22,20)(25,26,30,28)(27,32,29,31);(1,2)(3,5)(4,6)(7,8)(9,11,13,10)(12,14,16,15)(17,19)(18,23)(20,22)(21,24)(25,28,30,26)(27,31,29,32);(1,3,4,7)(2,5,6,8)(9,12,13,16)(10,14,11,15)(17,20,21,23)(18,24,22,19)(25,29,30,27)(26,31,28,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=h,c^2=fh,d^2=h,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dg,e^b=ef,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3140842);