Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2076493 |
| 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 | |
2259 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)(3,6)(4,7)(5,8)(9,10,14,12)(11,16,13,15)(17,18,23,24)(19,20,22,21)(26,30)(28,31);(1,3)(2,5)(4,8)(6,7)(9,11,14,13)(10,15,12,16)(17,19,23,22)(18,21,24,20)(25,26,29,30)(27,28,32,31);(3,8)(5,6)(9,10,14,12)(11,15,13,16)(17,20,23,21)(18,19,24,22)(25,27,29,32)(26,31,30,28);(1,2)(3,6)(4,7)(5,8)(9,12,14,10)(11,15,13,16)(17,21,23,20)(18,19,24,22)(25,28)(26,27)(29,31)(30,32);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,15,12,16)(17,22,23,19)(18,20,24,21)(25,27)(26,28)(29,32)(30,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=fghi,c^2=fghi,d^2=fgh,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2188088);