Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C25 |
| commutator, socle | C23 | C24×C4 |
| prime radical | C24 | C23×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1324 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,19)(13,16)(15,21)(17,24)(18,23)(20,22)(25,26,29,33,30,34,41,48)(27,31,37,44,36,43,50,54)(28,32,40,47,39,46,52,56)(35,42,49,53,38,45,51,55);(2,5)(3,7)(9,11,13,18)(10,14,16,23)(12,20,21,17)(15,24,19,22)(25,27)(26,31)(28,38)(29,37)(30,36)(32,45)(33,44)(34,43)(35,39)(40,51)(41,50)(42,46)(47,55)(48,54)(49,52)(53,56);(9,12,13,21)(10,15,16,19)(11,17,18,20)(14,22,23,24)(25,28)(26,32)(27,35)(29,40)(30,39)(31,42)(33,47)(34,46)(36,38)(37,49)(41,52)(43,45)(44,53)(48,56)(50,51)(54,55);(1,3,4,7)(2,6,5,8)(10,16)(14,23)(15,19)(22,24)(28,39)(32,46)(35,38)(40,52)(42,45)(47,56)(49,51)(53,55);(2,5)(6,8)(27,36)(31,43)(35,38)(37,50)(42,45)(44,54)(49,51)(53,55)Polycyclic presentation:
2^9|a^2=h,b^2=g,c^2=g,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cgi,d^b=df,e^b=ei,d^c=di92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6316039);