Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6533123 |
| 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 ◁ #2296 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1306 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)(9,10,13,16)(11,17,18,14)(12,15,20,22)(19,23,24,21)(25,26,30,34,29,33,42,48)(27,35,38,51,37,31,44,46)(28,39,41,47,40,49,43,32)(36,52,54,45,53,56,55,50);(2,5)(6,8)(9,11)(10,14)(12,19)(13,18)(15,21)(16,17)(20,24)(22,23)(25,27,29,37)(26,31,33,35)(28,36,40,53)(30,38,42,44)(32,45,47,50)(34,46,48,51)(39,56,49,52)(41,54,43,55);(11,18)(14,17)(19,24)(21,23)(25,28,29,40)(26,32,33,47)(27,36,37,53)(30,43,42,41)(31,45,35,50)(34,49,48,39)(38,55,44,54)(46,52,51,56);(1,3,4,7)(2,6,5,8)(12,20)(15,22)(19,24)(21,23)(26,33)(31,35)(32,47)(34,48)(39,49)(45,50)(46,51)(52,56);(9,12,13,20)(10,15,16,22)(11,19,18,24)(14,21,17,23)(26,33)(27,37)(32,47)(34,48)(36,53)(38,44)(39,49)(54,55)Polycyclic presentation:
2^9|a^2=fgi,b^2=i,c^2=i,d^2=fhi,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ei,c^b=ch,e^b=ei,g^c=gi,e^d=eh71 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6451751);