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 | C2×#121 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1358 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)(3,5)(4,6)(7,8)(9,10)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,29,33,30,34,41,48)(27,31,37,44,36,43,50,54)(28,32,39,46,40,47,52,56)(35,42,49,53,38,45,51,55);(1,3)(2,5)(4,7)(6,8)(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,27)(26,31)(28,35)(29,37)(30,36)(32,42)(33,44)(34,43)(38,40)(39,49)(41,50)(45,47)(46,53)(48,54)(51,52)(55,56);(2,6)(3,7)(25,28,29,39,30,40,41,52)(26,32,33,46,34,47,48,56)(27,35,37,49,36,38,50,51)(31,42,44,53,43,45,54,55);(2,6)(5,8)(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,27)(26,31)(28,38)(29,37)(30,36)(32,45)(33,44)(34,43)(35,40)(39,51)(41,50)(42,47)(46,55)(48,54)(49,52)(53,56);(10,16)(14,17)(15,24)(22,23)(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=f,c^2=h,d^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=df,e^b=ei,d^c=di,e^d=ei92 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6318932);