Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #6806623 |
| 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 ◁ #2396 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
1142 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)(3,5)(4,6)(7,8)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26,30,34)(27,31,36,42)(28,32,39,45)(29,33,40,46)(35,41,47,51)(37,43,49,53)(38,44,50,54)(48,52,55,56);(1,3,4,7)(2,5,6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(27,35)(28,38)(29,40)(31,41)(32,44)(33,46)(36,47)(37,49)(39,50)(42,51)(43,53)(45,54);(2,6)(5,8)(25,27,29,35,30,36,40,47)(26,31,33,41,34,42,46,51)(28,37,38,48,39,49,50,55)(32,43,44,52,45,53,54,56);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(27,36)(31,42)(35,47)(37,49)(41,51)(43,53)(48,55)(52,56);(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(25,28,29,38,30,39,40,50)(26,32,33,44,34,45,46,54)(27,37,35,48,36,49,47,55)(31,43,41,52,42,53,51,56)Polycyclic presentation:
2^9|a^2=fi,b^2=fg,c^2=h,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di74 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 6721422);