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 | |
1468 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,4)(3,6,8,7)(9,10)(11,12)(13,14)(15,16)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30)(33,34)(35,41)(36,44)(37,38)(39,42)(40,45)(43,47)(46,48);(1,3,5,8)(2,6,4,7)(9,10)(11,12)(13,14)(15,16)(18,19)(23,25)(24,26)(30,31)(34,38)(35,42)(36,45)(47,48);(2,4)(6,7)(10,14)(11,15)(17,19,22,18)(20,23,28,25)(21,24,29,26)(27,31,32,30)(33,35)(34,39)(36,46)(37,42)(38,41)(40,48)(43,45)(44,47);(1,4,5,2)(3,7,8,6)(9,11,13,15)(10,12,14,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,34)(35,39)(36,40)(37,38)(41,42)(43,47)(44,45)(46,48);(1,3)(2,6)(4,7)(5,8)(9,12)(10,15)(11,14)(13,16)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)(33,36)(34,40)(35,43)(37,45)(38,44)(39,47)(41,48)(42,46)Polycyclic presentation:
2^9|a^2=gh,b^2=ghi,c^2=i,d^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, 6176588);