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 | |
1350 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)(9,10,13,16)(11,14,17,21)(12,19,20,15)(18,24,23,22)(25,26,30,33)(27,35,36,50)(28,32,40,48)(29,42,44,34)(31,43,45,38)(37,51,39,53)(41,56,55,49)(46,54,47,52);(9,11)(10,14)(12,18)(13,17)(15,22)(16,21)(19,24)(20,23)(25,27,30,36)(26,31,33,45)(28,39,40,37)(29,43,44,38)(32,47,48,46)(34,50,42,35)(41,52,55,54)(49,51,56,53);(1,3,5,4)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(27,36)(28,40)(31,45)(32,48)(35,50)(38,43)(41,55)(49,56);(1,4,5,3)(2,6,7,8)(11,17)(14,21)(18,23)(22,24)(25,28,30,40)(26,32,33,48)(27,37,36,39)(29,41,44,55)(31,46,45,47)(34,49,42,56)(35,51,50,53)(38,52,43,54);(9,12,13,20)(10,15,16,19)(11,18,17,23)(14,22,21,24)(26,33)(28,40)(31,45)(34,42)(35,50)(37,39)(41,55)(52,54)Polycyclic presentation:
2^9|a^2=h,b^2=i,c^2=g,d^2=gi,e^2=hi,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6405367);