Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469971 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1336 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,13,15)(11,14,17,21)(12,19,20,16)(18,24,23,22)(25,26)(27,36)(28,33)(29,34)(30,45)(31,35)(32,39)(37,52)(38,47)(40,51)(41,55)(42,49)(43,50)(44,56)(46,54)(48,53);(9,11)(10,14)(12,18)(13,17)(15,21)(16,22)(19,24)(20,23)(25,27,31,40)(26,32,35,47)(28,41,43,37)(29,39,30,38)(33,48,50,46)(34,51,45,36)(42,53,44,54)(49,52,56,55);(1,3,4,8)(2,5,6,7)(10,15)(11,17)(16,19)(18,23)(25,28,31,43)(26,33,35,50)(27,37,40,41)(29,42,30,44)(32,46,47,48)(34,49,45,56)(36,52,51,55)(38,53,39,54);(11,17)(14,21)(18,23)(22,24)(25,29,31,30)(26,34,35,45)(27,38,40,39)(28,42,43,44)(32,36,47,51)(33,49,50,56)(37,53,41,54)(46,52,48,55);(9,12,13,20)(10,16,15,19)(11,18,17,23)(14,22,21,24)(26,35)(28,43)(32,47)(34,45)(36,51)(37,41)(42,44)(53,54)Polycyclic presentation:
2^9|a^2=g,b^2=i,c^2=gh,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406429);