Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of D32 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C16 | D16 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C2 | C8×D16 |
| prime radical | #2262 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C16 ◁ C2×C16 ◁ C4×C16 ◁ C8×C16 ◁ 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:
(1,2,6,12,7,14,31,44,9,15,32,45,8,33,28,13)(3,16,18,36,20,47,52,61,21,49,53,62,34,48,19,54)(4,11,22,41,23,42,30,57,25,43,29,59,24,55,5,26)(10,35,37,50,38,58,63,51,40,60,64,56,39,27,46,17);(1,3)(2,10)(4,17)(5,27)(6,18)(7,20)(8,34)(9,21)(11,36)(12,37)(13,46)(14,38)(15,40)(16,26)(19,28)(22,35)(23,50)(24,56)(25,51)(29,60)(30,58)(31,52)(32,53)(33,39)(41,47)(42,61)(43,62)(44,63)(45,64)(48,59)(49,57)(54,55);(1,4,6,22,7,23,31,30,9,25,32,29,8,24,28,5)(2,11,12,41,14,42,44,57,15,43,45,59,33,55,13,26)(3,17,18,35,20,50,52,58,21,51,53,60,34,56,19,27)(10,36,37,47,38,61,63,49,40,62,64,48,39,54,46,16)Polycyclic presentation:
2^9|a^2=e,c^2=e,d^2=fh,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,d^b=df,f^b=fh,h^b=hi152 conjugacy classes — view the full character table.
| maximal subgroup of | #72225308, #72225355, #72225356, … |
| maximal normal subgroup of | #72225308, #72225355, #72225356, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58277);