Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
1434 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,4)(3,7,5,8)(9,10,13,12)(11,14,15,16)(17,18)(19,23)(20,24)(21,22)(25,26,29,27)(28,31,32,30);(1,3,6,5)(2,7,4,8)(11,15)(14,16)(17,18,21,22)(19,23,24,20);(1,3,6,5)(2,8,4,7)(9,11,13,15)(10,14,12,16)(17,19,21,24)(18,20,22,23)(25,27,29,26)(28,31,32,30);(1,4,6,2)(3,7,5,8)(25,28)(26,30)(27,31)(29,32);(1,5,6,3)(2,7,4,8)(9,12,13,10)(11,16,15,14)(17,20)(18,19)(21,23)(22,24)Polycyclic presentation:
2^9|a^2=ghi,b^2=g,c^2=fghi,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2151701);