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 | C22×#41 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1260 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)(3,7)(4,5)(6,8)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26,30,34,29,33,42,48)(27,35,38,50,37,31,43,45)(28,39,41,53,40,32,46,47)(36,44,52,55,51,49,54,56);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,24)(18,19)(25,27)(26,31)(28,36)(29,37)(30,38)(32,44)(33,35)(34,45)(39,49)(40,51)(41,52)(42,43)(46,54)(47,55)(48,50)(53,56);(2,5)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,29,40)(26,32,33,39)(27,36,37,51)(30,41,42,46)(31,44,35,49)(34,47,48,53)(38,52,43,54)(45,55,50,56);(1,3,4,8)(2,6,5,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(26,33)(27,37)(28,40)(34,48)(38,43)(41,46)(44,49)(55,56);(10,18)(15,19)(16,17)(23,24)(26,33)(27,37)(32,39)(34,48)(36,51)(38,43)(47,53)(52,54)Polycyclic presentation:
2^9|a^2=h,c^2=f,d^2=fg,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, 6316561);