Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| automorphism | Aut(G) order 2097152 |
| 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 | |
1734 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(3,7)(9,10)(11,12)(13,15)(14,16)(17,18)(19,24)(20,22)(21,23)(25,26,30,27)(28,31,29,32);(2,5)(6,8)(9,11,13,16)(10,12,15,14)(17,19,22,21)(18,23,20,24)(26,27)(31,32);(9,11)(10,14)(12,15)(13,16)(17,20,22,18)(19,23,21,24)(25,27,30,26)(28,32,29,31);(1,2)(3,6)(4,5)(7,8)(25,28,30,29)(26,31,27,32);(1,3)(2,6)(4,7)(5,8)(9,12,13,14)(10,11,15,16)(17,21,22,19)(18,23,20,24)(25,29,30,28)(26,32,27,31)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=f,d^2=gh,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863085);