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) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1440 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,4,6)(3,5,7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,20)(21,24)(22,23)(25,26)(27,30)(28,33)(29,32)(31,35)(34,37)(36,39)(38,40);(9,11,13,16)(10,14,15,12)(17,18,21,24)(19,20,23,22)(26,27)(28,35)(29,37)(38,39);(11,16)(12,14)(19,23)(20,22)(25,27,30,26)(28,33,35,31)(29,32,37,34)(36,38,40,39);(1,2)(3,5)(4,6)(7,8)(9,11)(10,12)(13,16)(14,15)(17,19,21,23)(18,22,24,20)(25,28)(26,31)(27,33)(29,36)(30,35)(32,38)(34,39)(37,40);(1,3,4,7)(2,5,6,8)(9,12)(10,11)(13,14)(15,16)(17,20,21,22)(18,23,24,19)(25,29,30,37)(26,32,27,34)(28,36,35,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=g,d^2=fh,e^2=i,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.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5292102);