Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| 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 | C2×#121 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1231 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)(12,16)(13,15)(17,21)(18,20)(19,22)(23,24)(25,26,31,45,29,34,30,35)(27,36,46,48,38,32,39,52)(28,40,44,33,42,51,43,50)(37,53,55,49,41,56,54,47);(1,3,7,4)(2,5,8,6)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,29,38)(26,32,34,36)(28,41,42,37)(30,39,31,46)(33,49,50,47)(35,48,45,52)(40,53,51,56)(43,55,44,54);(1,4,7,3)(2,5,8,6)(10,15)(11,17)(12,19)(20,24)(25,28,29,42)(26,33,34,50)(27,37,38,41)(30,44,31,43)(32,47,36,49)(35,51,45,40)(39,55,46,54)(48,53,52,56);(1,5,7,6)(2,3,8,4)(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,20,21,24)(26,34)(27,38)(33,50)(35,45)(37,41)(39,46)(40,51)(54,55);(1,6,7,5)(2,3,8,4)(11,17)(14,21)(18,23)(20,24)Polycyclic presentation:
2^9|a^2=gh,b^2=f,c^2=f,d^2=fh,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855462);