Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5605735 |
| 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 | |
1636 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(2,4)(6,10)(7,13)(12,17)(15,19)(21,22,25,28)(23,29,30,26)(24,32,33,27)(31,34,36,35)(37,38)(39,44)(40,46)(41,45)(42,43)(47,49)(48,52)(50,51);(8,16)(12,17)(15,19)(18,20)(23,30)(24,33)(26,29)(27,32)(38,43)(39,46)(45,51)(47,52);(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18)(21,23)(22,26)(24,31)(25,30)(27,34)(28,29)(32,35)(33,36)(37,39,42,46)(38,44,43,40)(41,47,50,52)(45,49,51,48);(1,2)(3,4)(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,22)(23,29)(24,27)(25,28)(26,30)(31,35)(32,33)(34,36)(37,40,42,44)(38,39,43,46)(41,49,50,48)(45,52,51,47);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,24)(22,27)(23,31)(25,33)(26,34)(28,32)(29,35)(30,36)(37,41,42,50)(38,45,43,51)(39,47,46,52)(40,48,44,49)Polycyclic presentation:
2^9|a^2=fh,c^2=gh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=eg,d^c=dfg,e^d=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5526644);