Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #392 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4×C8 | C23 |
| commutator, socle, prime radical | C23 | C2×C4×C8 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
294 conjugacy classes of subgroups.
| minimal degree | μ(G) = 56 |
| transitive degree | μt(G) = 512 |
A minimal faithful permutation representation — 3 generators on 56 points:
(1,2)(3,6)(4,5)(7,8)(9,10,12,15,13,16,19,22)(11,14,18,21,17,20,23,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,42)(36,43)(37,41)(38,46)(39,44)(40,45)(47,53)(48,51)(49,52)(50,54)(55,56);(2,5)(6,8)(9,11)(10,14)(12,18)(13,17)(15,21)(16,20)(19,23)(22,24)(25,27,28,35,29,36,38,47,30,37,39,48,40,49,50,55)(26,31,33,42,34,43,46,53,32,41,44,51,45,52,54,56);(1,3,4,7)(2,6,5,8)(11,17)(14,20)(18,23)(21,24)(26,32)(31,41)(33,44)(34,45)(42,51)(43,52)(46,54)(53,56)Polycyclic presentation:
2^9|a^2=e,b^2=f,c^2=d,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,c^b=cg176 conjugacy classes — view the full character table.
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 35411);