Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| automorphism | Aut(G) order 4194304 |
| 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 | C22×#127 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2100 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,3)(6,7)(9,10)(11,15)(12,13)(14,16)(18,22)(20,23)(25,26,30,29)(27,31,32,28);(1,2)(3,5)(4,6)(7,8)(9,11,13,16)(10,14,12,15)(17,18,21,22)(19,20,24,23)(25,27,30,32)(26,28,29,31);(2,3)(4,8)(10,12)(14,15)(19,24)(20,23)(26,29)(27,32);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,19,21,24)(18,23,22,20)(25,28)(26,27)(29,32)(30,31);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,14,16,15)(17,20,21,23)(18,24,22,19)(25,29,30,26)(27,28,32,31)Polycyclic presentation:
2^9|a^2=hi,b^2=fi,d^2=fg,e^2=f,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, 5143672);