Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2258 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(3,7)(9,10)(11,16)(12,13)(14,15)(18,21)(19,23)(25,26)(27,30)(28,32)(29,31);(1,2)(3,8)(4,5)(6,7)(10,15)(13,16)(19,23)(22,24)(25,26,29,31)(27,28,32,30);(2,4)(6,8)(9,11,14,12)(10,13,15,16)(17,18)(19,22)(20,21)(23,24)(25,27,29,32)(26,30,31,28);(1,3)(2,6)(4,8)(5,7)(9,12,14,11)(10,16,15,13)(17,19)(18,22)(20,23)(21,24)(25,26)(27,30)(28,32)(29,31);(1,4,5,2)(3,6,7,8)(9,13,14,16)(10,12,15,11)(17,18)(19,22)(20,21)(23,24)(25,28,29,30)(26,27,31,32)Polycyclic presentation:
2^9|b^2=ghi,c^2=fgi,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3856104);