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)(8,15)(12,17)(13,19)(18,20)(21,22,26,23)(24,27,32,29)(25,30,34,28)(31,36,33,35)(38,42)(39,45)(40,48)(51,52);(6,10)(11,16)(12,17)(18,20)(22,23)(27,29)(28,30)(35,36)(37,38,41,42)(39,46,45,43)(40,44,48,50)(47,52,49,51);(5,6)(7,11)(8,12)(9,10)(13,18)(14,16)(15,17)(19,20)(21,23,26,22)(24,29,32,27)(25,30,34,28)(31,36,33,35)(37,38)(39,43)(40,44)(41,42)(45,46)(47,51)(48,50)(49,52);(1,2)(3,4)(5,7,9,14)(6,11,10,16)(8,13,15,19)(12,18,17,20)(21,24,26,32)(22,27,23,29)(25,33,34,31)(28,36,30,35)(37,39,41,45)(38,43,42,46)(40,49,48,47)(44,52,50,51);(5,8)(6,12)(7,13)(9,15)(10,17)(11,18)(14,19)(16,20)(21,25,26,34)(22,28,23,30)(24,31,32,33)(27,35,29,36)(37,40)(38,44)(39,47)(41,48)(42,50)(43,51)(45,49)(46,52)Polycyclic presentation:
2^9|a^2=fi,b^2=g,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, 4775752);