Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #6132643 |
| 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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 40 points:
(2,3)(6,7)(9,10)(11,14)(12,16)(13,15)(19,23)(22,24)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(1,2,5,3)(4,6,8,7)(9,11,13,16)(10,14,15,12)(18,21)(22,24)(27,34)(29,37)(31,33)(32,36);(2,3)(4,8)(17,18,20,21)(19,22,23,24)(25,27)(26,31)(28,33)(29,35)(30,34)(32,38)(36,39)(37,40);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(1,4,5,8)(2,6,3,7)(9,12)(10,11)(13,14)(15,16)(17,19)(18,22)(20,23)(21,24)(25,29,30,37)(26,32,28,36)(27,35,34,40)(31,38,33,39)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=fgh,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dfi,e^b=eg,d^c=dfg,e^c=ef62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6100875);