Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2401 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1185 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)(4,8)(5,6)(9,10)(11,17)(12,16)(13,15)(14,19)(18,24)(20,22)(21,23)(25,26,29,31)(27,35,37,32)(28,33,41,43)(30,34,40,44)(36,45,50,47)(38,49,52,46)(39,53,42,48)(51,55,54,56);(9,11,13,19)(10,14,15,17)(12,20,21,18)(16,24,23,22)(26,31)(27,36)(28,39)(30,40)(32,45)(33,48)(35,47)(37,50)(41,42)(43,53)(46,49)(51,54);(2,6)(7,8)(25,27,30,36,29,37,40,50)(26,32,34,47,31,35,44,45)(28,38,42,51,41,52,39,54)(33,46,48,56,43,49,53,55);(1,3,5,4)(2,7,6,8)(10,15)(14,17)(16,23)(22,24)(26,31)(27,37)(33,43)(34,44)(36,50)(38,52)(48,53)(51,54);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,15,23)(11,18,19,20)(14,22,17,24)(25,28,40,39,29,41,30,42)(26,33,44,53,31,43,34,48)(27,38,50,54,37,52,36,51)(32,46,45,55,35,49,47,56)Polycyclic presentation:
2^9|a^2=i,b^2=fi,c^2=h,d^2=gi,e^2=fghi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6750668);