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, cosocle | C24 | C25 |
| prime radical | C25 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
1656 conjugacy classes of subgroups.
| minimal degree | μ(G) = 52 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 52 points:
(1,2,3,4)(6,10)(11,16)(12,17)(18,20)(21,22,26,23)(24,29,32,27)(25,30,34,28)(31,35,33,36)(40,47)(43,50)(45,51)(48,52);(7,13)(11,16)(14,19)(18,20)(22,23)(27,29)(28,30)(35,36)(37,38,41,44)(39,42,46,49)(40,43,47,50)(45,48,51,52);(5,6,9,10)(7,11,13,16)(8,12,15,17)(14,18,19,20)(21,23,26,22)(24,29,32,27)(25,30,34,28)(31,36,33,35)(37,38)(39,42)(40,43)(41,44)(45,48)(46,49)(47,50)(51,52);(1,2)(3,4)(5,7)(6,11)(8,14)(9,13)(10,16)(12,18)(15,19)(17,20)(21,24,26,32)(22,27,23,29)(25,33,34,31)(28,36,30,35)(37,39,41,46)(38,42,44,49)(40,45,47,51)(43,48,50,52);(5,8,9,15)(6,12,10,17)(7,14,13,19)(11,18,16,20)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)(37,40)(38,43)(39,45)(41,47)(42,48)(44,50)(46,51)(49,52)Polycyclic presentation:
2^9|a^2=i,b^2=fh,c^2=fg,d^2=h,e^2=fg,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, 4775910);