Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470185 |
| 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 | C22×D4 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2293 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1347 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26)(27,32)(28,34)(29,35)(30,33)(31,36)(37,45)(38,44)(39,47)(40,50)(41,48)(42,46)(43,49)(51,56)(52,55)(53,54);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,39)(26,32,36,47)(28,37,40,51)(29,42,30,38)(33,44,35,46)(34,45,50,56)(41,53,43,52)(48,54,49,55);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(26,33)(28,40)(29,30)(32,44)(34,49)(35,36)(37,51)(38,42)(45,55)(46,47)(48,50)(54,56);(11,18)(14,22)(17,19)(21,23)(25,28,31,40)(26,34,36,50)(27,37,39,51)(29,43,30,41)(32,45,47,56)(33,48,35,49)(38,53,42,52)(44,54,46,55);(1,3,4,7)(2,6,5,8)(10,16)(14,22)(15,24)(21,23)(25,29,31,30)(26,35,36,33)(27,38,39,42)(28,41,40,43)(32,44,47,46)(34,48,50,49)(37,52,51,53)(45,55,56,54)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=h,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=dh,e^b=ei,d^c=di,e^c=ei,g^c=gi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6386698);