Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7324500 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1279 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(9,10,13,16)(11,17,18,14)(12,20,21,15)(19,22,24,23)(25,26,31,35)(27,36,37,48)(28,33,40,49)(29,34,30,45)(32,43,46,39)(38,53,41,56)(42,51,44,52)(47,54,50,55);(1,2)(3,7)(4,6)(5,8)(9,11,13,18)(10,14,16,17)(12,19,21,24)(15,22,20,23)(27,37)(28,40)(32,46)(33,49)(36,48)(39,43)(42,44)(51,52);(1,3)(2,5)(4,8)(6,7)(25,27,30,39,31,37,29,43)(26,32,34,36,35,46,45,48)(28,41,42,54,40,38,44,55)(33,50,52,56,49,47,51,53);(2,6)(3,8)(11,18)(14,17)(19,24)(22,23)(25,28,31,40)(26,33,35,49)(27,38,37,41)(29,44,30,42)(32,47,46,50)(34,52,45,51)(36,53,48,56)(39,55,43,54);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,19,18,24)(14,22,17,23)(25,29,31,30)(26,34,35,45)(27,39,37,43)(28,42,40,44)(32,48,46,36)(33,51,49,52)(38,54,41,55)(47,53,50,56)Polycyclic presentation:
2^9|a^2=fi,b^2=f,c^2=g,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=ehi,d^c=dh,e^c=ei,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7240579);