Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | |
1318 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)(25,26,31,35)(27,32,37,46)(28,40,42,48)(29,34,30,43)(33,44,49,41)(36,50,51,56)(38,47,39,53)(45,54,55,52);(2,6)(7,8)(9,10,13,16)(11,17,19,14)(12,15,20,22)(18,23,24,21)(25,27,31,37)(26,32,35,46)(28,36,42,51)(29,38,30,39)(33,45,49,55)(34,47,43,53)(40,50,48,56)(41,52,44,54);(9,11,13,19)(10,14,16,17)(12,18,20,24)(15,21,22,23)(25,28)(26,33)(27,36)(29,41)(30,44)(31,42)(32,45)(34,48)(35,49)(37,51)(38,52)(39,54)(40,43)(46,55)(47,56)(50,53);(1,3,5,4)(2,7,6,8)(9,12)(10,15)(11,18)(13,20)(14,21)(16,22)(17,23)(19,24)(27,37)(32,46)(36,51)(38,39)(45,55)(47,53)(50,56)(52,54);(1,4,5,3)(2,8,6,7)(12,20)(15,22)(18,24)(21,23)(25,29,31,30)(26,34,35,43)(27,38,37,39)(28,41,42,44)(32,47,46,53)(33,48,49,40)(36,52,51,54)(45,56,55,50)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6479372);