Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469995 |
| 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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1324 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)(11,15)(12,16)(13,17)(14,18)(19,22)(20,23)(21,24)(25,26,29,34,30,33,44,50)(27,35,38,45,36,51,43,31)(28,39,42,55,41,32,47,49)(37,53,54,46,40,52,56,48);(9,11,14,21)(10,15,18,24)(12,20,13,19)(16,23,17,22)(25,27,30,36)(26,31,33,45)(28,40,41,37)(29,43,44,38)(32,48,39,46)(34,51,50,35)(42,54,47,56)(49,52,55,53);(2,5)(6,8)(9,12,14,13)(10,16,18,17)(11,19,21,20)(15,22,24,23)(27,36)(31,45)(35,51)(37,40)(38,43)(46,48)(52,53)(54,56);(9,13,14,12)(10,17,18,16)(11,20,21,19)(15,23,24,22)(25,28,30,41)(26,32,33,39)(27,37,36,40)(29,42,44,47)(31,46,45,48)(34,49,50,55)(35,52,51,53)(38,54,43,56);(1,3,4,7)(2,6,5,8)(10,18)(15,24)(16,17)(22,23)(26,33)(28,41)(31,45)(34,50)(35,51)(37,40)(42,47)(54,56)Polycyclic presentation:
2^9|a^2=f,b^2=h,c^2=hi,d^2=h,e^2=g,f^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=dh,f^b=fi,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6397713);