Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6996375 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,6)(3,7,8,5)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,36,31,35,45,46)(27,37,39,48,28,33,41,32)(29,34,43,52,44,51,53,56)(38,50,42,55,40,49,54,47);(1,3,4,8)(2,5,6,7)(11,17)(12,20)(14,21)(15,19)(25,27)(26,32)(28,31)(29,42)(30,41)(33,36)(34,50)(35,48)(37,46)(38,43)(39,45)(40,53)(44,54)(47,52)(49,51)(55,56);(2,6)(5,7)(9,11,13,17)(10,14,16,21)(12,18,20,23)(15,22,19,24)(25,28,31,27)(26,33,35,37)(29,40,44,38)(30,41,45,39)(32,46,48,36)(34,49,51,50)(42,43,54,53)(47,56,55,52);(3,8)(5,7)(11,17)(14,21)(18,23)(22,24)(25,29,30,43,31,44,45,53)(26,34,36,52,35,51,46,56)(27,38,39,42,28,40,41,54)(32,47,37,50,48,55,33,49);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,35)(27,28)(34,51)(36,46)(38,40)(39,41)(42,54)(52,56)Polycyclic presentation:
2^9|a^2=fhi,b^2=g,c^2=h,d^2=fg,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,f^b=fi,d^c=dhi,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7105028);