Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6432843 |
| 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)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,21)(18,22)(19,23)(20,24)(25,26,30,35)(27,37,40,32)(28,42,45,52)(29,36,31,34)(33,46,51,44)(38,53,43,56)(39,48,41,49)(47,54,50,55);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,19,20,17)(15,23,24,21)(25,27)(26,32)(28,43)(29,39)(30,40)(31,41)(33,50)(34,48)(35,37)(36,49)(38,45)(42,53)(44,55)(46,54)(47,51)(52,56);(9,12,13,20)(10,15,16,24)(11,17,18,19)(14,21,22,23)(25,28,30,45)(26,33,35,51)(27,38,40,43)(29,44,31,46)(32,47,37,50)(34,42,36,52)(39,54,41,55)(48,56,49,53);(10,16)(14,22)(15,24)(21,23)(25,29,30,31)(26,34,35,36)(27,39,40,41)(28,44,45,46)(32,48,37,49)(33,42,51,52)(38,54,43,55)(47,56,50,53);(1,3,4,7)(2,6,5,8)(27,40)(32,37)(38,43)(39,41)(47,50)(48,49)(53,56)(54,55)Polycyclic presentation:
2^9|a^2=i,b^2=hi,c^2=h,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,e^b=ei,g^c=gi74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6342128);