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) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | #14715 | C2 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3758 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)(3,6,7,8)(9,10,14,16)(11,12,13,15)(17,18)(19,24)(20,22)(21,23)(25,26,29,32)(27,30,28,31);(2,5)(6,8)(9,11,14,13)(10,15,16,12)(17,19)(18,20)(21,22)(23,24);(3,7)(6,8)(10,16)(11,13)(19,22)(20,24)(25,27,29,28)(26,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,18)(19,24)(20,22)(21,23)(25,26)(27,30)(28,31)(29,32);(1,3,4,7)(2,6,5,8)(9,13,14,11)(10,12,16,15)(17,20,21,24)(18,22,23,19)(25,28,29,27)(26,31,32,30)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=fghi,e^2=ghi,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.
| maximal subgroup of | #72232582, #72232616, #72232691, … |
| maximal normal subgroup of | #72232582, #72232616, #72232691, … |
| maximal quotient of | #72331298 |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2450141);