Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #509 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | C2×D8 |
| commutator | C8 | C22×C16 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C2 | C16×D8 |
| prime radical | #508 | C8 |
| LCS | G ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C32 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
137 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,8,15,35,49,9,14,36,48,32,12,44,46,10,16,37,50,5,29,31,56,7,38,34,59,33,47,61,64)(3,17,19,23,21,41,54,25,20,51,53,58,18,11,40,4,22,24,55,26,30,42,60,27,39,45,62,57,52,43,63,28);(1,3)(2,11)(4,13)(5,30)(6,19)(7,39)(8,21)(9,20)(10,22)(12,45)(14,42)(15,24)(16,43)(17,29)(18,32)(23,56)(25,59)(26,49)(27,48)(28,50)(31,60)(33,52)(34,62)(35,54)(36,53)(37,55)(38,41)(40,44)(46,57)(47,51)(58,64)(61,63);(1,4,6,24,8,26,35,42,9,27,36,45,32,57,44,43,10,28,37,17,5,23,31,41,7,25,34,51,33,58,61,11)(2,3,13,19,15,21,49,54,14,20,48,53,12,18,46,40,16,22,50,55,29,30,56,60,38,39,59,62,47,52,64,63)Polycyclic presentation:
2^9|a^2=e,c^2=e,d^2=fi,e^2=g,f^2=i,g^2=h,h^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fi,d^b=df,f^b=fi128 conjugacy classes — view the full character table.
| maximal subgroup of | #72225353, #72225369, #72225565, … |
| maximal normal subgroup of | #72225353, #72225369, #72225565, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58269);