Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2760365 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2500 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,7)(4,8)(5,6)(11,15)(14,16)(17,18)(19,23)(20,22)(21,24)(26,29)(27,31);(1,3)(2,4)(5,8)(6,7)(9,10,13,12)(11,14,15,16)(17,19,22,21)(18,23,20,24)(27,31)(30,32);(2,6)(3,8)(17,18,22,20)(19,24,21,23)(25,26)(27,30)(28,29)(31,32);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,18)(19,23,21,24)(25,27)(26,30)(28,31)(29,32);(1,4,5,7)(2,8,6,3)(9,12,13,10)(11,16,15,14)(17,21,22,19)(18,23,20,24)(25,26)(27,30)(28,29)(31,32)Polycyclic presentation:
2^9|b^2=ghi,c^2=fh,d^2=fh,e^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,d^c=df,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2712067);