Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806839 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2402 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,8)(5,6)(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,32)(28,40)(29,34)(30,46)(31,36)(33,45)(35,38)(37,53)(39,48)(41,52)(42,50)(43,56)(44,49)(47,55)(51,54);(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27,31,39)(26,32,36,48)(28,41,45,44)(29,38,42,30)(33,49,40,52)(34,35,50,46)(37,54,55,43)(47,56,53,51);(2,6)(7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,42,54,31,45,29,43)(26,33,50,56,36,40,34,51)(27,37,38,44,39,55,30,41)(32,47,35,52,48,53,46,49);(1,3,5,4)(2,7,6,8)(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(25,29,31,42)(26,34,36,50)(27,38,39,30)(28,43,45,54)(32,35,48,46)(33,51,40,56)(37,44,55,41)(47,52,53,49);(1,4,5,3)(2,8,6,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,30,29,39,31,38,42,27)(26,35,34,32,36,46,50,48)(28,44,43,37,45,41,54,55)(33,49,51,53,40,52,56,47)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=fhi,d^2=fg,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,e^c=ei,e^d=ei68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6753758);