Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| 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 | |
1512 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)(6,7)(9,10)(11,16)(12,15)(13,14)(17,18)(19,24)(20,22)(21,23)(25,26,28,30)(27,31,32,29);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18,21,23)(19,20,22,24);(9,12)(10,15)(11,13)(14,16)(17,19,21,22)(18,20,23,24);(1,2)(3,7)(4,5)(6,8)(9,12)(10,11)(13,15)(14,16)(17,18)(19,20)(21,23)(22,24)(25,27,28,32)(26,29,30,31);(1,3,4,8)(2,6,5,7)(9,13,14,10)(11,16,15,12)(17,20)(18,22)(19,23)(21,24)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=g,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4801787);