Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6470922 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1226 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)(3,8)(4,6)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,34)(27,37,39,49)(28,42,44,51)(29,36,30,35)(32,41,48,40)(33,45,52,46)(38,47,43,50)(53,54,56,55);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,17)(15,23)(16,21)(18,20)(22,24)(25,27,29,40,31,39,30,41)(26,32,35,37,34,48,36,49)(28,43,45,55,44,38,46,54)(33,50,51,56,52,47,42,53);(11,17)(14,21)(18,19)(22,23)(25,28,29,45,31,44,30,46)(26,33,35,51,34,52,36,42)(27,38,40,54,39,43,41,55)(32,47,37,53,48,50,49,56);(9,12,13,20)(10,15,16,24)(11,18,17,19)(14,22,21,23)(27,39)(32,48)(37,49)(38,43)(40,41)(47,50)(53,56)(54,55);(1,4,5,3)(2,6,7,8)(10,16)(14,21)(15,24)(22,23)(26,34)(28,44)(32,48)(35,36)(37,49)(38,43)(45,46)(54,55)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=gi,d^2=hi,e^2=fg,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,g^a=gi,c^b=ch,d^b=dh,e^c=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6394362);