Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5473759 |
| 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 | |
1627 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)(23,27)(24,31)(25,26)(28,32)(29,35)(30,33)(34,36)(38,42)(39,45)(44,48)(47,51);(8,16)(12,17)(15,19)(18,20)(22,25)(27,33)(28,31)(34,35)(37,38,41,42)(39,43,45,46)(40,44,49,48)(47,50,51,52);(5,6,9,10)(7,14,13,11)(8,12,16,17)(15,20,19,18)(21,23)(22,27)(24,29)(25,33)(26,30)(28,34)(31,35)(32,36)(37,38)(39,46)(40,48)(41,42)(43,45)(44,49)(47,50)(51,52);(1,2)(3,4)(5,7,9,13)(6,11,10,14)(8,15,16,19)(12,18,17,20)(21,24,26,32)(22,28,25,31)(23,29,30,36)(27,34,33,35)(37,39)(38,43)(40,47)(41,45)(42,46)(44,50)(48,52)(49,51);(5,8)(6,12)(7,15)(9,16)(10,17)(11,18)(13,19)(14,20)(21,25,26,22)(23,27,30,33)(24,31,32,28)(29,34,36,35)(37,40,41,49)(38,44,42,48)(39,47,45,51)(43,50,46,52)Polycyclic presentation:
2^9|b^2=h,c^2=gh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 5382783);