Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #5956507 |
| automorphism | Aut(G) order 1048576 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26 |
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 | |
1311 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,6)(3,4,8,7)(11,17)(12,19)(14,20)(15,22)(27,33)(31,36)(34,39)(38,40)(41,42,45,46)(43,47,49,53)(44,51,52,48)(50,56,55,54);(1,2,5,6)(3,7,8,4)(11,17)(14,20)(18,23)(21,24)(26,30)(27,33)(32,37)(34,39)(42,46)(43,49)(44,52)(54,56);(2,6)(3,8)(9,10,13,16)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,29,30)(27,31,33,36)(28,32,35,37)(34,38,39,40)(41,42)(43,47)(44,48)(45,46)(49,53)(50,54)(51,52)(55,56);(1,3)(2,4)(5,8)(6,7)(9,11,13,17)(10,14,16,20)(12,18,19,23)(15,21,22,24)(25,27)(26,31)(28,34)(29,33)(30,36)(32,38)(35,39)(37,40)(41,43)(42,47)(44,50)(45,49)(46,53)(48,54)(51,56)(52,55);(1,4)(2,3)(5,7)(6,8)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(25,28,29,35)(26,32,30,37)(27,34,33,39)(31,38,36,40)(41,44,45,52)(42,48,46,51)(43,50,49,55)(47,54,53,56)Polycyclic presentation:
2^9|a^2=g,b^2=fg,c^2=i,d^2=ghi,e^2=fgh,b^a=bf,c^a=cg,d^a=di,e^a=ehi,c^b=ch,d^b=dgi,e^b=ef,d^c=dfg62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6018137);