Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4465744 |
| automorphism | Aut(G) order 2097152 |
| 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 | #15257 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2517 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,6)(4,5)(7,8)(10,13)(11,15)(17,18,22,24)(19,21,20,23)(25,26,31,32)(27,28,29,30);(2,5)(6,8)(9,10)(11,16)(12,13)(14,15)(17,19)(18,23)(20,22)(21,24)(25,27)(26,30)(28,32)(29,31);(3,7)(6,8)(10,13)(14,16)(18,24)(19,20)(25,28,31,30)(26,27,32,29);(1,3,4,7)(2,6,5,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,19)(18,21,24,23)(25,29,31,27)(26,28,32,30);(1,2)(3,6)(4,5)(7,8)(9,11,12,15)(10,14,13,16)(17,21)(18,19)(20,24)(22,23)(25,30,31,28)(26,27,32,29)Polycyclic presentation:
2^9|a^2=g,c^2=hi,d^2=fgi,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^b=ef,d^c=df68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4433438);