Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1225 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,5,7)(3,6,4,8)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,34)(27,31,37,46)(28,32,39,48)(29,33,43,52)(35,44,53,55)(36,45,41,50)(38,47,54,56)(40,49,42,51);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27)(26,31)(28,38)(29,41)(30,37)(32,47)(33,50)(34,46)(35,40)(36,43)(39,54)(42,53)(44,49)(45,52)(48,56)(51,55);(2,7)(6,8)(25,28,30,39)(26,32,34,48)(27,35,37,53)(29,42,43,40)(31,44,46,55)(33,51,52,49)(36,54,41,38)(45,56,50,47);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(28,39)(32,48)(35,53)(38,54)(40,42)(44,55)(47,56)(49,51);(1,4,5,3)(2,8,7,6)(11,19)(14,17)(18,24)(22,23)(27,36)(29,43)(31,45)(33,52)(35,54)(37,41)(38,53)(40,42)(44,56)(46,50)(47,55)(49,51)Polycyclic presentation:
2^9|a^2=fgi,b^2=g,c^2=i,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6626694);