Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| 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 ◁ C4×#117 ◁ 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)(3,7)(4,5)(6,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,21)(19,22)(23,24)(25,26,31,46,30,35,29,34)(27,36,45,32,39,52,38,48)(28,40,42,51,44,33,43,50)(37,54,55,49,41,53,56,47);(2,5)(6,7)(9,11)(10,14)(12,19)(13,18)(15,17)(16,22)(20,23)(21,24)(25,27,30,39)(26,32,35,48)(28,41,44,37)(29,45,31,38)(33,49,40,47)(34,52,46,36)(42,55,43,56)(50,54,51,53);(1,3,4,8)(2,6,5,7)(10,15)(11,18)(16,20)(19,24)(25,28)(26,33)(27,37)(29,43)(30,44)(31,42)(32,47)(34,51)(35,40)(36,53)(38,56)(39,41)(45,55)(46,50)(48,49)(52,54);(11,18)(14,17)(19,24)(22,23)(25,29,30,31)(26,34,35,46)(27,38,39,45)(28,42,44,43)(32,36,48,52)(33,50,40,51)(37,55,41,56)(47,54,49,53);(9,12,13,21)(10,16,15,20)(11,19,18,24)(14,22,17,23)(26,35)(32,48)(33,40)(34,46)(36,52)(47,49)(50,51)(53,54)Polycyclic presentation:
2^9|a^2=fg,b^2=i,c^2=gh,d^2=i,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6406483);