Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5955303 |
| 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 | |
1642 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)(3,4)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,18,22,21)(19,24,23,20)(27,33)(28,35)(30,36)(31,38);(1,3,6,5)(2,7,8,4)(10,11)(15,16)(17,19)(18,20)(21,24)(22,23)(27,33)(30,36)(34,39)(37,40);(1,4,6,7)(2,3,8,5)(9,11,14,10)(12,16,13,15)(17,20,22,24)(18,23,21,19)(25,26,29,32)(27,30,33,36)(28,31,35,38)(34,37,39,40);(9,12,14,13)(10,15,11,16)(17,18,22,21)(19,20,23,24)(25,27)(26,30)(28,34)(29,33)(31,37)(32,36)(35,39)(38,40);(1,5,6,3)(2,7,8,4)(9,13,14,12)(10,16,11,15)(17,21,22,18)(19,24,23,20)(25,28)(26,31)(27,34)(29,35)(30,37)(32,38)(33,39)(36,40)Polycyclic presentation:
2^9|a^2=fh,b^2=fg,c^2=fi,d^2=g,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,d^b=di104 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5863196);