Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #13657 |
| automorphism | ? |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #2177 |
| commutator | C16 | C22×C8 |
| Frattini, cosocle | C4×C16 | C23 |
| socle | C2 | #12045 |
| prime radical | #2262 | C4 |
| LCS | G ▷ C16 ▷ C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ C4×C16 ◁ #1311 ◁ G | |
| LECS | G ▷ C4×C16 ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
133 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,8,13,25,43,9,15,33,47,32,14,7,35)(3,16,19,50,21,52,36,51,23,53,20,60,22,61,54,38)(4,24,26,62,28,11,5,30,29,44,34,63,59,45,27,49)(10,37,39,58,41,56,40,18,42,31,46,17,48,55,64,57);(1,3)(2,10)(4,17)(5,31)(6,20)(7,36)(8,21)(9,23)(11,38)(12,46)(13,41)(14,48)(15,42)(16,49)(18,59)(19,33)(22,32)(24,60)(25,54)(26,56)(27,37)(28,57)(29,58)(30,53)(34,55)(35,40)(39,47)(43,64)(44,50)(45,51)(52,62)(61,63);(1,4,25,5,32,59,6,26,9,29,7,27,8,28,33,34)(2,11,35,62,14,24,47,49,15,45,43,63,13,44,12,30)(3,17,54,31,22,18,20,56,23,58,36,37,21,57,19,55)(10,38,40,52,48,60,39,16,42,51,64,61,41,50,46,53)Polycyclic presentation:
2^9|a^2=e,c^2=eh,d^2=fhi,e^2=g,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fh,h^a=hi,d^b=dfi,e^b=ei,f^b=fh,h^b=hi,d^c=di68 conjugacy classes — view the full character table.
| maximal subgroup of | #72224647, #72224652, #72224654, … |
| maximal normal subgroup of | #72224647, #72224652, #72224654, … |
| maximal quotient of | #72230700, C2×#151086, C3×#151086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 58282);