Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6885065 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C23×D4 |
| commutator, Frattini, cosocle | C22×C4 | C25 |
| prime radical | #2407 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #2290 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1409 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 56 points:
(2,5)(3,7)(9,10,13,16)(11,17,19,14)(12,20,21,15)(18,22,24,23)(25,26,30,34)(27,36,38,32)(28,40,41,49)(29,35,31,44)(33,45,50,42)(37,52,54,56)(39,48,43,47)(46,55,51,53);(1,2,4,5)(3,6,7,8)(9,11,13,19)(10,14,16,17)(12,18,21,24)(15,22,20,23)(25,27,30,38)(26,32,34,36)(28,37,41,54)(29,43,31,39)(33,46,50,51)(35,48,44,47)(40,56,49,52)(42,55,45,53);(2,5)(6,8)(12,21)(15,20)(18,24)(22,23)(25,28)(26,33)(27,37)(29,42)(30,41)(31,45)(32,46)(34,50)(35,49)(36,51)(38,54)(39,53)(40,44)(43,55)(47,56)(48,52);(1,3,4,7)(2,6,5,8)(9,12,13,21)(10,15,16,20)(11,18,19,24)(14,22,17,23)(26,34)(27,38)(28,41)(35,44)(39,43)(42,45)(46,51)(52,56);(11,19)(14,17)(18,24)(22,23)(25,29,30,31)(26,35,34,44)(27,39,38,43)(28,42,41,45)(32,47,36,48)(33,49,50,40)(37,53,54,55)(46,56,51,52)Polycyclic presentation:
2^9|a^2=fh,b^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=dfh,g^c=gi65 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6824360);