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 | |
1438 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,3)(4,7,8,6)(9,10,13,14)(11,15,16,12)(17,18,21,19)(20,23,24,22)(25,26)(27,33)(28,30)(29,32)(31,34)(35,40)(36,37)(38,39);(1,2,5,3)(4,7,8,6)(10,14)(11,16)(17,19,21,18)(20,23,24,22)(26,28)(27,34)(29,37)(39,40);(2,3)(6,7)(9,11,13,16)(10,15,14,12)(25,27)(26,31)(28,33)(29,38)(30,34)(32,40)(35,37)(36,39);(1,3,5,2)(4,7,8,6)(9,11)(10,15)(12,14)(13,16)(17,20)(18,22)(19,23)(21,24)(25,28,30,26)(27,33,34,31)(29,36,37,32)(35,40,38,39);(1,4,5,8)(2,6,3,7)(9,12,13,15)(10,11,14,16)(25,29,30,37)(26,32,28,36)(27,35,34,38)(31,39,33,40)Polycyclic presentation:
2^9|a^2=hi,b^2=gi,c^2=gh,d^2=g,e^2=h,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, 5151003);