Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6907993 |
| 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 | C2×#121 | D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#529 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1396 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)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,35)(27,36,31,32)(28,39,42,46)(29,43,44,34)(33,37,49,40)(38,47,45,50)(41,54,55,52)(48,53,56,51);(1,3)(2,5)(4,8)(6,7)(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,30,31)(26,32,35,36)(28,40,42,37)(29,38,44,45)(33,39,49,46)(34,47,43,50)(41,53,55,51)(48,52,56,54);(3,8)(5,7)(12,21)(15,20)(18,24)(22,23)(25,28,31,40,30,42,27,37)(26,33,36,39,35,49,32,46)(29,41,45,53,44,55,38,51)(34,48,50,52,43,56,47,54);(2,6)(5,7)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(25,29,30,44)(26,34,35,43)(27,38,31,45)(28,41,42,55)(32,47,36,50)(33,48,49,56)(37,51,40,53)(39,52,46,54);(11,19)(14,17)(18,24)(22,23)(29,44)(34,43)(38,45)(41,55)(47,50)(48,56)(51,53)(52,54)Polycyclic presentation:
2^9|a^2=fhi,b^2=i,c^2=g,d^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,d^c=dfh,e^d=ei62 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6855759);