Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6667580 |
| 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 | C22×D4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2399 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1321 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)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,29,33)(27,35,38,31)(28,40,42,32)(30,34,44,50)(36,46,54,53)(37,47,39,51)(41,48,55,56)(43,52,45,49);(9,11)(10,14)(12,18)(13,19)(15,22)(16,17)(20,23)(21,24)(25,27,29,38)(26,31,33,35)(28,41,42,55)(30,37,44,39)(32,48,40,56)(34,51,50,47)(36,45,54,43)(46,52,53,49);(2,5)(6,8)(25,28,29,42)(26,32,33,40)(27,36,38,54)(30,45,44,43)(31,46,35,53)(34,52,50,49)(37,41,39,55)(47,56,51,48);(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,33)(28,42)(31,35)(34,50)(36,54)(41,55)(43,45)(47,51);(1,3,4,7)(2,6,5,8)(11,19)(14,17)(18,24)(22,23)(26,33)(27,37)(30,44)(31,47)(32,40)(35,51)(36,41)(38,39)(43,45)(46,56)(48,53)(54,55)Polycyclic presentation:
2^9|a^2=f,b^2=i,c^2=i,d^2=fi,e^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi68 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6619903);