Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C25×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
3199 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)(9,10,12,14)(11,15,16,13)(17,18,21,19)(20,23,24,22)(25,26,29,30)(27,32,28,31);(3,7)(6,8)(11,16)(13,15)(17,18,21,19)(20,23,24,22)(26,30)(27,28);(2,5)(6,8)(18,19)(22,23)(25,27,29,28)(26,31,30,32);(1,2)(3,6)(4,5)(7,8)(9,11)(10,13)(12,16)(14,15)(17,19,21,18)(20,23,24,22)(25,28,29,27)(26,31,30,32);(1,3)(2,6)(4,7)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,22)(19,23)(21,24)(25,26)(27,32)(28,31)(29,30)Polycyclic presentation:
2^9|a^2=hi,b^2=fh,c^2=f,d^2=h,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
| maximal subgroup of | #72262814, #72262817, #72262829, … |
| maximal normal subgroup of | #72262814, #72262817, #72262829, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6079840);