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 | |
1452 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 48 points:
(2,5)(6,8)(9,10,12,15)(11,13,16,14)(17,18,21,22)(19,23,25,29)(20,27,28,24)(26,32,31,30)(33,34)(35,39)(36,44)(37,38)(40,45)(41,42)(43,48)(46,47);(2,5)(3,7)(9,11,12,16)(10,13,15,14)(18,22)(19,25)(20,28)(30,32)(34,38)(35,41)(40,44)(43,47);(1,2,4,5)(3,6,7,8)(9,11,12,16)(10,14,15,13)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31)(33,34)(35,42)(36,44)(37,38)(39,41)(40,45)(43,46)(47,48);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16)(17,20,21,28)(18,24,22,27)(19,26,25,31)(23,30,29,32)(33,36,37,45)(34,40,38,44)(35,43,41,47)(39,46,42,48)Polycyclic presentation:
2^9|a^2=fgh,b^2=fhi,c^2=i,e^2=f,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, 6176408);