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, prime radical, cosocle | C24 | C25 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1558 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(8,15)(12,17)(13,19)(18,20)(21,22)(23,26)(24,27)(25,30)(28,34)(29,31)(32,36)(33,35)(37,38,41,44)(39,45,47,42)(40,48,49,43)(46,50,52,51);(6,10)(11,16)(12,17)(18,20)(22,23)(24,31)(28,30)(32,33)(37,38,41,44)(39,45,47,42)(40,43,49,48)(46,51,52,50);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,23,26,22)(24,29,31,27)(25,30,34,28)(32,36,33,35)(37,38)(39,42)(40,43)(41,44)(45,47)(46,50)(48,49)(51,52);(1,2)(3,4)(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,24)(22,27)(23,29)(25,33)(26,31)(28,36)(30,35)(32,34)(37,39,41,47)(38,42,44,45)(40,46,49,52)(43,50,48,51);(5,8)(6,12)(7,13)(9,15)(10,17)(11,18)(14,19)(16,20)(21,25,26,34)(22,28,23,30)(24,32,31,33)(27,35,29,36)(37,40,41,49)(38,43,44,48)(39,46,47,52)(42,50,45,51)Polycyclic presentation:
2^9|a^2=i,b^2=fg,c^2=f,d^2=fh,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, 4775698);