Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3332717 |
| 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 | C26 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2374 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(10,11)(12,16)(17,18)(19,22)(20,24)(21,23)(26,29)(30,31);(1,2)(3,6)(4,5)(7,8)(10,11)(14,15)(17,19,21,24)(18,22,23,20)(25,26)(27,30)(28,29)(31,32);(2,5)(6,7)(9,10,13,11)(12,14,16,15)(17,20)(18,19)(21,22)(23,24)(25,26,28,29)(27,30,32,31);(1,3)(2,6)(4,8)(5,7)(9,11,13,10)(12,15,16,14)(17,19)(18,20)(21,24)(22,23)(25,27,28,32)(26,30,29,31);(1,2)(3,7)(4,5)(6,8)(9,12)(10,14)(11,15)(13,16)(17,20)(18,19)(21,22)(23,24)(25,26)(27,31)(28,29)(30,32)Polycyclic presentation:
2^9|a^2=fgi,b^2=gh,c^2=fghi,d^2=fghi,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef,e^d=ef74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 3461334);