Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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, cosocle | C24 | C25 |
| prime radical | #15263 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2361 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,12)(13,15)(14,16)(18,19)(20,24)(25,26,29,28)(27,31,32,30);(2,5)(6,8)(9,11)(10,14)(12,15)(13,16)(17,18,21,19)(20,22,24,23)(25,27,29,32)(26,30,28,31);(3,7)(6,8)(10,15)(12,14)(17,18,21,19)(20,23,24,22)(25,27)(26,31)(28,30)(29,32);(1,2)(3,8)(4,5)(6,7)(9,11)(10,12)(13,16)(14,15)(17,19,21,18)(20,22,24,23)(25,27)(26,31)(28,30)(29,32);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,15)(13,14)(17,20)(18,22)(19,23)(21,24)(25,28,29,26)(27,31,32,30)Polycyclic presentation:
2^9|a^2=fg,b^2=fgi,c^2=fghi,d^2=fghi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3477352);