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 | |
1450 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,4,6)(3,7,8,5)(9,10,13,14)(11,12,15,16)(20,27)(24,29)(25,31)(30,32)(34,38)(35,41)(40,43)(42,47);(3,8)(5,7)(10,14)(11,15)(18,22)(23,28)(24,29)(30,32)(33,34,37,38)(35,39,41,45)(36,43,44,40)(42,48,47,46);(9,11,13,15)(10,12,14,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,28)(27,29)(31,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,3)(2,5)(4,8)(6,7)(9,11)(10,12)(13,15)(14,16)(17,19,21,26)(18,23,22,28)(20,25,27,31)(24,30,29,32)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,2)(3,5)(4,6)(7,8)(9,12)(10,11)(13,16)(14,15)(17,20,21,27)(18,24,22,29)(19,25,26,31)(23,30,28,32)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48)Polycyclic presentation:
2^9|a^2=fi,b^2=g,c^2=fg,d^2=fgh,e^2=fgh,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, 6091696);