Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533195 |
| 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 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1327 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)(11,14,17,20)(12,15,19,22)(18,21,23,24)(25,26,30,35)(27,36,38,32)(28,41,43,48)(29,34,31,44)(33,45,49,42)(37,52,53,56)(39,51,40,47)(46,54,50,55);(1,3,5,4)(2,6,8,7)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,30,38)(26,32,35,36)(28,37,43,53)(29,39,31,40)(33,46,49,50)(34,47,44,51)(41,56,48,52)(42,54,45,55);(1,3,5,4)(2,7,8,6)(11,17)(14,20)(18,23)(21,24)(25,28,30,43)(26,33,35,49)(27,37,38,53)(29,42,31,45)(32,46,36,50)(34,48,44,41)(39,54,40,55)(47,52,51,56);(1,4,5,3)(2,6,8,7)(9,12)(10,15)(11,18)(13,19)(14,21)(16,22)(17,23)(20,24)(27,38)(32,36)(37,53)(39,40)(46,50)(47,51)(52,56)(54,55);(12,19)(15,22)(18,23)(21,24)(25,29,30,31)(26,34,35,44)(27,39,38,40)(28,42,43,45)(32,47,36,51)(33,48,49,41)(37,54,53,55)(46,52,50,56)Polycyclic presentation:
2^9|a^2=hi,b^2=f,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,g^c=gi,e^d=eh77 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6473815);