Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6884841 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | C2×#449 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2298 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1392 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)(4,6)(5,8)(10,14)(12,19)(15,21)(17,20)(25,26,31,36)(27,32,41,49)(28,34,43,50)(29,35,30,45)(33,44,51,42)(37,52,38,47)(39,48,40,55)(46,54,56,53);(1,3,4,8)(2,5,6,7)(9,10,13,14)(11,15,18,21)(12,16,19,22)(17,23,20,24)(25,27)(26,32)(28,37)(29,39)(30,40)(31,41)(33,46)(34,52)(35,48)(36,49)(38,43)(42,53)(44,54)(45,55)(47,50)(51,56);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28)(26,33)(27,37)(29,42)(30,44)(31,43)(32,46)(34,45)(35,50)(36,51)(38,41)(39,53)(40,54)(47,48)(49,56)(52,55);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(25,28)(26,34)(27,38)(29,44)(30,42)(31,43)(32,47)(33,45)(35,51)(36,50)(37,41)(39,53)(40,54)(46,48)(49,52)(55,56);(10,14)(15,21)(16,22)(23,24)(25,29,31,30)(26,35,36,45)(27,39,41,40)(28,42,43,44)(32,48,49,55)(33,50,51,34)(37,53,38,54)(46,47,56,52)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=fh,d^2=fh,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi,e^d=ei65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6861270);