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 | |
1453 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)(3,6)(4,8)(5,7)(9,10,12,14)(11,13,15,16)(18,22)(23,25)(24,29)(30,31)(33,34,38,39)(35,40,41,36)(37,42,46,47)(43,48,45,44);(1,3,5,8)(2,6,7,4)(11,15)(13,16)(20,28)(24,29)(26,32)(30,31)(34,39)(35,41)(42,47)(43,45);(9,11)(10,13)(12,15)(14,16)(17,18,21,22)(19,25,27,23)(20,24,28,29)(26,31,32,30)(33,35)(34,40)(36,39)(37,45)(38,41)(42,44)(43,46)(47,48);(1,3)(2,4)(5,8)(6,7)(17,19,21,27)(18,23,22,25)(20,26,28,32)(24,30,29,31)(33,36,38,40)(34,41,39,35)(37,44,46,48)(42,45,47,43);(1,4)(2,3)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,28)(18,24,22,29)(19,26,27,32)(23,30,25,31)(33,37)(34,42)(35,43)(36,44)(38,46)(39,47)(40,48)(41,45)Polycyclic presentation:
2^9|a^2=h,b^2=fi,c^2=g,d^2=fg,e^2=g,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, 6098419);