Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2805613 |
| 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 | |
1546 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 40 points:
(1,2,5,4)(3,6,7,8)(10,11)(14,15)(17,18,22,21)(19,23,20,24)(25,26)(27,33)(28,30)(29,36)(31,34)(32,37)(35,38)(39,40);(2,4)(6,8)(9,10,13,11)(12,15,16,14)(17,18)(19,24)(20,23)(21,22)(26,28)(27,34)(29,37)(38,39);(17,19,22,20)(18,23,21,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,3,5,7)(2,6,4,8)(9,11,13,10)(12,14,16,15)(17,20,22,19)(18,23,21,24)(25,28,30,26)(27,33,34,31)(29,32,37,36)(35,38,40,39);(1,4,5,2)(3,6,7,8)(9,12,13,16)(10,14,11,15)(17,21,22,18)(19,24,20,23)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)Polycyclic presentation:
2^9|a^2=fghi,b^2=hi,c^2=gh,d^2=f,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef80 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 2750178);