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 | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1958 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)(3,7)(4,5)(6,8)(10,11)(12,16)(17,18,22,20)(19,23,21,24)(26,30)(28,31);(2,5)(6,7)(10,11)(14,15)(17,19,22,21)(18,23,20,24)(25,26,29,30)(27,28,32,31);(2,5)(6,7)(9,10,13,11)(12,14,16,15)(17,19,22,21)(18,24,20,23)(25,27,29,32)(26,31,30,28);(1,3,4,8)(2,6,5,7)(9,11,13,10)(12,15,16,14)(17,20,22,18)(19,23,21,24)(25,26)(27,31)(28,32)(29,30);(1,2)(3,7)(4,5)(6,8)(9,12,13,16)(10,14,11,15)(17,21,22,19)(18,23,20,24)(25,28,29,31)(26,27,30,32)Polycyclic presentation:
2^9|a^2=fgh,b^2=gi,c^2=fg,d^2=g,e^2=fg,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, 3783774);