Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433049 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1356 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,16)(13,15)(17,20)(18,21)(19,22)(23,24)(25,26,31,36)(27,37,40,49)(28,42,46,51)(29,35,30,34)(32,41,48,39)(33,45,52,44)(38,47,43,50)(53,55,56,54);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(25,27,30,41,31,40,29,39)(26,32,35,49,36,48,34,37)(28,43,44,55,46,38,45,54)(33,50,42,53,52,47,51,56);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(25,28,29,45,31,46,30,44)(26,33,34,51,36,52,35,42)(27,38,39,55,40,43,41,54)(32,47,37,53,48,50,49,56);(10,15)(14,20)(16,22)(21,24)(25,29,31,30)(26,34,36,35)(27,39,40,41)(28,44,46,45)(32,37,48,49)(33,42,52,51)(38,54,43,55)(47,56,50,53);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(27,40)(32,48)(37,49)(38,43)(39,41)(47,50)(53,56)(54,55)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=f,d^2=i,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,g^a=gi,c^b=ch,e^b=ei,d^c=di68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344370);