Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3858029 |
| 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 | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1950 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,14)(12,13)(15,16)(17,18,22,23)(19,21,20,24)(25,26,30,28)(27,29,32,31);(1,3)(2,6)(4,5)(7,8)(9,10,12,13)(11,14,16,15)(18,23)(21,24)(25,27,30,32)(26,31,28,29);(3,4)(6,7)(10,13)(14,15)(17,19,22,20)(18,24,23,21)(25,26,30,28)(27,29,32,31);(1,2)(3,7)(4,6)(5,8)(9,11)(10,14)(12,16)(13,15)(17,20,22,19)(18,21,23,24)(25,28,30,26)(27,31,32,29);(1,4,5,3)(2,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,21)(18,20)(19,23)(22,24)(25,29,30,31)(26,27,28,32)Polycyclic presentation:
2^9|a^2=gh,b^2=fi,c^2=fgi,d^2=fgi,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efg,c^b=ch,d^b=dfg,e^b=ef,e^c=ef,e^d=ef86 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3936431);