Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C24×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C25 |
| prime radical | C23×D4 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1881 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(2,5)(6,7)(10,14)(12,15)(17,18,21,23)(19,20,24,22)(25,26,28,30)(27,29,31,32);(1,2,4,5)(3,7,8,6)(9,10,13,14)(11,12,16,15)(17,19)(18,22)(20,23)(21,24)(27,31)(29,32);(2,5)(6,7)(9,11,13,16)(10,15,14,12)(17,18,21,23)(19,22,24,20)(25,26)(27,32)(28,30)(29,31);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(17,18)(19,20)(21,23)(22,24)(25,27,28,31)(26,29,30,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,15)(14,16)(17,20)(18,19)(21,22)(23,24)Polycyclic presentation:
2^9|a^2=fgh,b^2=h,c^2=hi,d^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5926643);