Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4417862 |
| 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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1630 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,4)(3,8)(11,15)(14,16)(17,18,20,22)(19,21,23,24)(25,26,29,32)(27,30,28,31);(1,2,5,4)(3,7,8,6)(10,13)(14,16)(19,23)(21,24)(25,27,29,28)(26,30,32,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16)(17,19,20,23)(18,21,22,24)(25,26)(27,30)(28,31)(29,32);(1,3)(2,7)(4,6)(5,8)(17,18)(19,21)(20,22)(23,24)(25,28,29,27)(26,31,32,30);(1,4,5,2)(3,7,8,6)(9,11)(10,14)(12,15)(13,16)(17,18)(19,21)(20,22)(23,24)Polycyclic presentation:
2^9|a^2=gi,b^2=g,c^2=fi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4339402);