Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ 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)(3,7)(4,8)(5,6)(25,26,30,35)(27,37,39,32)(28,42,44,50)(29,36,31,34)(33,45,49,43)(38,52,53,56)(40,47,41,48)(46,54,51,55);(2,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,27,30,39)(26,32,35,37)(28,38,44,53)(29,40,31,41)(33,46,49,51)(34,47,36,48)(42,56,50,52)(43,54,45,55);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28)(26,33)(27,38)(29,43)(30,44)(31,45)(32,46)(34,42)(35,49)(36,50)(37,51)(39,53)(40,54)(41,55)(47,56)(48,52);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(27,39)(32,37)(38,53)(40,41)(46,51)(47,48)(52,56)(54,55);(1,4,5,3)(2,8,6,7)(12,20)(15,22)(18,24)(21,23)(25,29,30,31)(26,34,35,36)(27,40,39,41)(28,43,44,45)(32,47,37,48)(33,42,49,50)(38,54,53,55)(46,56,51,52)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479405);