Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6433057 |
| 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 ◁ C4×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1272 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)(25,26,30,34)(27,35,36,48)(28,40,41,32)(29,43,37,33)(31,44,45,39)(38,52,50,49)(42,47,51,55)(46,53,54,56);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,20)(16,21)(19,23)(22,24)(27,36)(28,41)(31,45)(32,40)(35,48)(39,44)(42,51)(47,55);(11,17)(14,20)(18,23)(21,24)(25,27,37,44,30,36,29,39)(26,31,43,48,34,45,33,35)(28,38,51,56,41,50,42,53)(32,46,55,52,40,54,47,49);(1,3,4,7)(2,6,5,8)(10,15)(14,20)(16,22)(21,24)(25,28,30,41)(26,32,34,40)(27,38,36,50)(29,42,37,51)(31,46,45,54)(33,47,43,55)(35,49,48,52)(39,53,44,56);(9,12,13,19)(10,16,15,22)(11,18,17,23)(14,21,20,24)(28,41)(32,40)(38,50)(42,51)(46,54)(47,55)(49,52)(53,56)Polycyclic presentation:
2^9|a^2=i,c^2=gi,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,d^a=dh,g^a=gi,c^b=ch,d^b=di,e^d=ei74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6344561);