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) = C25×C4 |
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 | |
2546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(10,12)(14,15)(17,18,21,20)(19,23,24,22)(25,26)(27,31)(28,32)(29,30);(1,2,4,6)(3,7,8,5)(9,10)(11,15)(12,13)(14,16)(17,18)(19,22)(20,21)(23,24)(26,29)(28,31);(3,8)(5,7)(10,12)(14,15)(18,20)(22,23)(25,27,30,32)(26,31,29,28);(1,2)(3,7)(4,6)(5,8)(9,11,13,16)(10,14,12,15)(17,19,21,24)(18,22,20,23)(25,28)(26,27)(29,32)(30,31);(1,3,4,8)(2,5,6,7)(9,12,13,10)(11,14,16,15)(17,20,21,18)(19,22,24,23)(25,29,30,26)(27,31,32,28)Polycyclic presentation:
2^9|a^2=g,b^2=fghi,c^2=gi,d^2=h,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, 3856119);