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, prime radical | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2295 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1290 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2)(3,7)(4,6)(5,8)(10,14)(15,17)(16,21)(22,23)(25,26,31,36)(27,32,38,46)(28,41,43,50)(29,35,30,34)(33,44,49,42)(37,51,52,56)(39,48,40,47)(45,54,55,53);(1,3,5,4)(2,6,8,7)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22)(25,27)(26,32)(28,37)(29,39)(30,40)(31,38)(33,45)(34,47)(35,48)(36,46)(41,51)(42,53)(43,52)(44,54)(49,55)(50,56);(1,3,5,4)(2,7,8,6)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23)(25,28)(26,33)(27,37)(29,42)(30,44)(31,43)(32,45)(34,41)(35,50)(36,49)(38,52)(39,53)(40,54)(46,55)(47,51)(48,56);(1,4,5,3)(2,6,8,7)(12,20)(16,21)(18,24)(22,23)(27,38)(32,46)(37,52)(39,40)(45,55)(47,48)(51,56)(53,54);(9,12,13,20)(10,16,14,21)(11,18,19,24)(15,22,17,23)(25,29,31,30)(26,34,36,35)(27,39,38,40)(28,42,43,44)(32,47,46,48)(33,41,49,50)(37,53,52,54)(45,51,55,56)Polycyclic presentation:
2^9|a^2=i,b^2=f,c^2=f,d^2=fh,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ei,g^a=gi,c^b=ch,d^b=di,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, 6448419);