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 | |
1625 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)(5,6,10,8)(7,11,13,14)(9,16,17,12)(15,20,19,18)(21,22)(23,28)(24,30)(25,29)(26,27)(31,33)(32,36)(34,35)(39,45)(41,49)(43,46)(44,48);(7,13)(11,14)(15,19)(18,20)(22,27)(23,30)(29,35)(31,36)(37,38,42,40)(39,46,45,43)(41,48,49,44)(47,50,51,52);(5,7,10,13)(6,11,8,14)(9,15,17,19)(12,18,16,20)(21,23,26,30)(22,28,27,24)(25,31,34,36)(29,33,35,32)(37,39,42,45)(38,43,40,46)(41,47,49,51)(44,50,48,52);(1,2)(3,4)(5,8,10,6)(7,14,13,11)(9,16,17,12)(15,20,19,18)(21,24,26,28)(22,23,27,30)(25,33,34,32)(29,36,35,31)(37,40,42,38)(39,43,45,46)(41,48,49,44)(47,50,51,52);(5,9)(6,12)(7,15)(8,16)(10,17)(11,18)(13,19)(14,20)(21,25,26,34)(22,29,27,35)(23,31,30,36)(24,32,28,33)(37,41,42,49)(38,44,40,48)(39,47,45,51)(43,50,46,52)Polycyclic presentation:
2^9|a^2=fgh,b^2=g,c^2=h,d^2=h,e^2=fg,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, 5526868);