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 | #15263 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2483 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,6,7,8)(9,10,13,16)(11,15,12,14)(17,18,21,20)(19,23,24,22)(25,26,29,27)(28,31,32,30);(2,4)(6,8)(9,11)(10,14)(12,13)(15,16)(17,18)(19,22)(20,21)(23,24)(26,27)(28,32);(2,4)(6,8)(11,12)(14,15)(18,20)(22,23)(25,26,29,27)(28,31,32,30);(1,3)(2,6)(4,8)(5,7)(9,10)(11,15)(12,14)(13,16)(17,19)(18,22)(20,23)(21,24)(25,27,29,26)(28,30,32,31);(1,4,5,2)(3,6,7,8)(9,12,13,11)(10,15,16,14)(17,20,21,18)(19,22,24,23)(25,28,29,32)(26,30,27,31)Polycyclic presentation:
2^9|a^2=f,c^2=hi,d^2=hi,e^2=f,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, 3908698);