Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1302 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)(11,15)(12,16)(13,14)(17,20)(18,22)(19,21)(23,24)(25,26,31,45,30,34,29,35)(27,36,38,48,39,32,40,47)(28,41,42,33,43,50,44,49)(37,52,55,51,54,56,53,46);(2,6)(7,8)(10,14)(11,17)(16,21)(18,23)(25,27,30,39)(26,32,34,36)(28,37,43,54)(29,40,31,38)(33,46,49,51)(35,48,45,47)(41,56,50,52)(42,55,44,53);(1,3,5,4)(2,7,6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28)(26,33)(27,37)(29,42)(30,43)(31,44)(32,46)(34,49)(35,50)(36,51)(38,53)(39,54)(40,55)(41,45)(47,56)(48,52);(1,4,5,3)(2,8,6,7)(12,19)(16,21)(18,23)(22,24)(26,34)(32,36)(33,49)(35,45)(41,50)(46,51)(47,48)(52,56);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,31)(26,35,34,45)(27,38,39,40)(28,42,43,44)(32,47,36,48)(33,50,49,41)(37,53,54,55)(46,56,51,52)Polycyclic presentation:
2^9|a^2=gi,b^2=i,c^2=fi,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6457900);