Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| 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 | |
1496 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)(11,15)(14,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,34)(35,40)(36,42)(37,39)(38,45)(41,46)(43,48)(44,47);(3,8)(5,7)(10,13)(14,16)(18,22)(20,28)(23,25)(26,32)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,3)(2,5)(4,8)(6,7)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,15)(13,16)(17,20)(18,24)(19,26)(21,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,46)(34,41,37,45)(35,43,36,44)(40,47,42,48)Polycyclic presentation:
2^9|a^2=fgi,b^2=fg,c^2=fg,d^2=g,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5895689);