Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7309080 |
| 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 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1304 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(1,2,4,5)(3,6,7,8)(10,14)(11,17)(12,20)(22,24)(25,26,31,33,29,35,30,36)(27,37,44,48,39,32,40,46)(28,41,45,50,42,34,43,49)(38,47,54,55,52,51,53,56);(2,5)(3,7)(9,10,13,14)(11,18,17,15)(12,21,20,16)(19,22,23,24)(25,27,29,39)(26,32,35,37)(28,38,42,52)(30,40,31,44)(33,46,36,48)(34,47,41,51)(43,53,45,54)(49,55,50,56);(9,11)(10,15)(12,19)(13,17)(14,18)(16,22)(20,23)(21,24)(26,33)(30,31)(32,46)(34,49)(35,36)(37,48)(40,44)(41,50)(43,45)(47,55)(51,56)(53,54);(1,3,4,7)(2,6,5,8)(11,17)(15,18)(19,23)(22,24)(25,28,29,42)(26,34,35,41)(27,38,39,52)(30,43,31,45)(32,47,37,51)(33,49,36,50)(40,53,44,54)(46,55,48,56);(2,5)(6,8)(9,12,13,20)(10,16,14,21)(11,19,17,23)(15,22,18,24)(26,35)(28,42)(32,37)(33,36)(38,52)(43,45)(46,48)(53,54)Polycyclic presentation:
2^9|a^2=fg,b^2=hi,d^2=fh,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dfhi,e^b=eh,d^c=dh,g^c=gi,e^d=ei59 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 7405923);