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 | |
1652 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)(23,29)(24,30)(25,28)(26,27)(31,34)(32,36)(33,35)(37,38,41,44)(39,42,46,50)(40,47,48,43)(45,51,52,49);(8,16)(12,17)(15,19)(18,20)(22,27)(23,30)(25,33)(32,34)(37,39,41,46)(38,42,44,50)(40,45,48,52)(43,49,47,51);(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18)(21,23)(22,24)(25,31)(26,30)(27,29)(28,32)(33,36)(34,35)(37,39)(38,42)(40,45)(41,46)(43,49)(44,50)(47,51)(48,52);(1,2)(3,4)(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,24)(22,23)(25,34)(26,29)(27,30)(28,36)(31,35)(32,33)(37,38)(39,42)(40,43)(41,44)(45,49)(46,50)(47,48)(51,52);(5,8,9,16)(6,12,10,17)(7,15,13,19)(11,18,14,20)(21,25,26,33)(22,28,27,35)(23,31,30,36)(24,32,29,34)(37,40,41,48)(38,43,44,47)(39,45,46,52)(42,49,50,51)Polycyclic presentation:
2^9|a^2=fh,b^2=fh,c^2=fg,d^2=fg,e^2=fgh,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, 5533314);