Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4861280 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
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 | |
1629 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(6,10)(11,16)(12,17)(18,20)(21,22)(23,29)(24,31)(25,28)(26,27)(30,33)(32,36)(34,35)(38,42)(39,45)(40,48)(51,52);(7,13)(11,16)(14,19)(18,20)(22,27)(24,29)(28,35)(32,33)(37,38,41,42)(39,46,45,43)(40,44,48,50)(47,52,49,51);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23)(22,24)(25,30)(26,31)(27,29)(28,32)(33,35)(34,36)(37,38)(39,43)(40,44)(41,42)(45,46)(47,51)(48,50)(49,52);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24)(22,23)(25,33)(26,29)(27,31)(28,36)(30,35)(32,34)(37,39,41,45)(38,43,42,46)(40,49,48,47)(44,52,50,51);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,26,34)(22,28,27,35)(23,30,31,36)(24,32,29,33)(37,40,41,48)(38,44,42,50)(39,47,45,49)(43,51,46,52)Polycyclic presentation:
2^9|a^2=fi,b^2=h,c^2=fg,d^2=h,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4775906);