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 | C24 | C22×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1309 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)(25,26,31,34,30,35,29,43)(27,36,38,47,39,32,44,48)(28,40,41,50,42,49,45,33)(37,52,55,46,54,56,53,51);(2,6)(7,8)(10,14)(11,17)(16,21)(18,23)(25,27)(26,32)(28,37)(29,44)(30,39)(31,38)(33,46)(34,48)(35,36)(40,56)(41,55)(42,54)(43,47)(45,53)(49,52)(50,51);(1,3,5,4)(2,7,6,8)(9,11,13,17)(10,15,14,20)(12,18,19,23)(16,22,21,24)(25,28,30,42)(26,33,35,50)(27,37,39,54)(29,41,31,45)(32,46,36,51)(34,49,43,40)(38,53,44,55)(47,56,48,52);(1,4,5,3)(2,8,6,7)(12,19)(16,21)(18,23)(22,24);(9,12,13,19)(10,16,14,21)(11,18,17,23)(15,22,20,24)(25,29,30,31)(26,34,35,43)(27,38,39,44)(28,41,42,45)(32,47,36,48)(33,49,50,40)(37,53,54,55)(46,56,51,52)Polycyclic presentation:
2^9|a^2=g,c^2=f,d^2=fhi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=ei,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, 6457904);