Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6469956 |
| 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 ◁ C2×#376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1328 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)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,31,35)(27,36,39,47)(28,40,41,33)(29,34,30,44)(32,45,48,38)(37,51,52,50)(42,55,43,49)(46,53,54,56);(1,3,5,4)(2,6,7,8)(9,11)(10,14)(12,19)(13,18)(15,23)(16,22)(17,20)(21,24)(25,27,31,39)(26,32,35,48)(28,37,41,52)(29,38,30,45)(33,46,40,54)(34,47,44,36)(42,53,43,56)(49,51,55,50);(2,7)(6,8)(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,31,41)(26,33,35,40)(27,37,39,52)(29,42,30,43)(32,46,48,54)(34,49,44,55)(36,50,47,51)(38,53,45,56);(1,4,5,3)(2,8,7,6)(11,18)(14,22)(17,19)(21,23)(28,41)(33,40)(37,52)(42,43)(46,54)(49,55)(50,51)(53,56);(10,16)(14,22)(15,24)(21,23)(25,29,31,30)(26,34,35,44)(27,38,39,45)(28,42,41,43)(32,47,48,36)(33,49,40,55)(37,53,52,56)(46,51,54,50)Polycyclic presentation:
2^9|a^2=i,b^2=g,c^2=hi,d^2=gi,e^2=i,f^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fi,c^b=ch,d^b=dh,f^b=fi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6387063);