Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D16 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C16 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C32 | 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 ◁ C32 ◁ C2×C32 ◁ C4×C32 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
139 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,5,15,28,49,8,14,33,48,30,47,60,53,9,16,34,18,31,50,61,55,7,35,32,54,29,12,44,46)(3,17,20,51,22,39,57,25,21,52,56,58,41,40,63,4,11,24,42,23,43,38,64,26,36,45,62,59,19,10,37,27);(1,3)(2,10)(4,18)(5,22)(6,20)(7,36)(8,21)(9,11)(12,45)(13,27)(14,39)(15,17)(16,40)(19,29)(23,55)(24,50)(25,48)(26,54)(28,57)(30,41)(31,43)(32,62)(33,56)(34,42)(35,38)(37,44)(46,59)(47,52)(49,51)(53,58)(60,63)(61,64);(1,4,6,24,5,23,28,38,8,26,33,45,30,59,60,10,9,27,34,17,31,51,61,39,7,25,32,52,29,58,44,40)(2,11,13,42,15,43,49,64,14,36,48,62,47,19,53,37,16,3,18,20,50,22,55,57,35,21,54,56,12,41,46,63)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,d^a=df,f^a=fi,d^b=df,f^b=fi176 conjugacy classes — view the full character table.
| maximal subgroup of | #72225353, #72225355, #72225356, … |
| maximal normal subgroup of | #72225353, #72225355, #72225356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58268);