Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309060 |
| 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 | C22×D8 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1325 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(10,14)(11,17)(12,20)(22,24)(25,26,29,34,30,35,42,32)(27,31,37,45,38,46,53,43)(28,39,41,47,40,33,48,49)(36,50,52,54,51,44,55,56);(2,5)(3,7)(9,10,13,14)(11,18,17,15)(12,21,20,16)(19,22,23,24)(25,27,30,38)(26,31,35,46)(28,36,40,51)(29,37,42,53)(32,43,34,45)(33,44,39,50)(41,52,48,55)(47,54,49,56);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(26,32)(29,42)(31,43)(33,47)(34,35)(37,53)(39,49)(41,48)(44,54)(45,46)(50,56)(52,55);(1,3,4,7)(2,6,5,8)(11,17)(15,18)(19,23)(22,24)(25,28,30,40)(26,33,35,39)(27,36,38,51)(29,41,42,48)(31,44,46,50)(32,47,34,49)(37,52,53,55)(43,54,45,56);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(28,40)(33,39)(36,51)(41,48)(44,50)(47,49)(52,55)(54,56)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7405727);