Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #4597856 |
| 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 | |
1609 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)(5,6,9,12)(7,10,13,17)(8,15,16,11)(14,20,19,18)(21,22,26,27)(23,28,30,24)(25,29,34,35)(31,36,33,32)(37,38,42,45)(39,43,40,46)(41,44,49,51)(47,50,48,52);(7,13)(10,17)(14,19)(18,20)(22,27)(23,30)(29,35)(31,33)(37,39,42,40)(38,43,45,46)(41,47,49,48)(44,50,51,52);(5,7)(6,10)(8,14)(9,13)(11,18)(12,17)(15,20)(16,19)(21,23)(22,28)(24,27)(25,33)(26,30)(29,32)(31,34)(35,36)(37,40,42,39)(38,43,45,46)(41,48,49,47)(44,50,51,52);(1,2)(3,4)(5,6)(7,10)(8,11)(9,12)(13,17)(14,18)(15,16)(19,20)(21,24,26,28)(22,23,27,30)(25,32,34,36)(29,31,35,33)(37,38)(39,46)(40,43)(41,44)(42,45)(47,52)(48,50)(49,51);(5,8,9,16)(6,11,12,15)(7,14,13,19)(10,18,17,20)(21,25,26,34)(22,29,27,35)(23,31,30,33)(24,32,28,36)(37,41,42,49)(38,44,45,51)(39,47,40,48)(43,50,46,52)Polycyclic presentation:
2^9|a^2=gh,b^2=g,c^2=g,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,d^b=dg,e^c=ef68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 4511172);