Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | C22×#120 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
2297 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)(11,15)(12,13)(14,16)(17,18)(19,20)(21,23)(22,24)(25,26)(27,30)(28,29)(31,32);(3,7)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19,22,21)(18,23,24,20)(25,27)(26,30)(28,31)(29,32);(2,5)(6,8)(10,13)(14,15)(19,21)(20,23)(25,27,29,32)(26,31,28,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,23)(18,19,24,21)(25,27)(26,31)(28,30)(29,32);(1,3,4,7)(2,6,5,8)(9,11,12,16)(10,14,13,15)(17,21,22,19)(18,23,24,20)(25,28,29,26)(27,30,32,31)Polycyclic presentation:
2^9|a^2=fgh,b^2=f,c^2=fghi,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
| maximal subgroup of | #72219214, #72219215, #72219270, … |
| maximal normal subgroup of | #72219214, #72219215, #72219270, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5157402);