Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5211343 |
| 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 | |
1599 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)(7,13)(8,15)(11,16)(12,17)(21,22,26,23)(24,29,31,27)(25,28,33,30)(32,35,36,34)(37,38)(39,44)(40,46)(41,43)(42,45)(47,51)(48,52)(49,50);(6,10)(11,16)(12,17)(18,20)(22,23)(24,31)(28,30)(32,36)(37,39,43,40)(38,44,41,46)(42,48,50,47)(45,52,49,51);(5,6)(7,11)(8,12)(9,10)(13,16)(14,18)(15,17)(19,20)(21,23,26,22)(24,27,31,29)(25,28,33,30)(32,35,36,34)(37,40,43,39)(38,44,41,46)(42,48,50,47)(45,51,49,52);(1,2)(3,4)(5,7,9,13)(6,11,10,16)(8,14,15,19)(12,18,17,20)(21,24,26,31)(22,27,23,29)(25,32,33,36)(28,34,30,35)(37,41,43,38)(39,46,40,44)(42,49,50,45)(47,52,48,51);(5,8)(6,12)(7,14)(9,15)(10,17)(11,18)(13,19)(16,20)(21,25,26,33)(22,28,23,30)(24,32,31,36)(27,34,29,35)(37,42,43,50)(38,45,41,49)(39,47,40,48)(44,51,46,52)Polycyclic presentation:
2^9|a^2=f,b^2=g,c^2=fg,d^2=gh,e^2=fg,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=df,e^b=eg,d^c=dfg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5123256);