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 | |
1373 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)(3,4)(5,6)(7,8)(11,15)(14,16)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31)(33,34)(35,42)(36,38)(37,45)(39,41)(40,47)(43,46)(44,48);(2,6)(4,7)(10,13)(11,15)(20,27)(23,25)(28,32)(30,31)(33,35,39,36)(34,38,41,42)(37,44,47,43)(40,48,45,46);(1,2,5,6)(3,7,8,4)(9,10)(11,14)(12,13)(15,16)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30)(33,36,39,35)(34,38,41,42)(37,43,47,44)(40,48,45,46);(1,3)(2,7)(4,6)(5,8)(9,11)(10,14)(12,15)(13,16)(17,20,22,27)(18,23,19,25)(21,28,29,32)(24,30,26,31)(33,37,39,47)(34,40,41,45)(35,43,36,44)(38,46,42,48);(1,4,5,7)(2,3,6,8)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38)(34,35)(36,41)(37,46)(39,42)(40,43)(44,45)(47,48)Polycyclic presentation:
2^9|a^2=fghi,b^2=fg,c^2=ghi,d^2=hi,e^2=g,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, 6029300);