Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | #16284 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2091 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,7)(3,5,4,8)(9,10)(11,13)(12,14)(15,16)(17,18,21,23)(19,20,24,22)(25,26,31,32)(27,28,29,30);(1,3,6,4)(2,5,7,8)(10,11)(12,16)(17,18,21,23)(19,20,24,22)(25,27,31,29)(26,28,32,30);(2,7)(3,4)(9,10,13,11)(12,15,16,14)(17,19,21,24)(18,22,23,20)(25,28,31,30)(26,27,32,29);(1,4,6,3)(2,5,7,8)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,29,31,27)(26,28,32,30);(1,5)(2,3)(4,7)(6,8)(9,12,13,16)(10,14,11,15)(17,20)(18,19)(21,22)(23,24)(25,30,31,28)(26,27,32,29)Polycyclic presentation:
2^9|a^2=fg,b^2=fg,c^2=fhi,d^2=i,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5238226);