Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #127312 |
| automorphism | ? |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×H(3,ℤ/4) |
| commutator | C2×C8 | C2×C42 |
| Frattini, cosocle | C2×#131 | C23 |
| socle | C2 | C2×#1289 |
| prime radical | C22×#41 | D4 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C2×#131 ▷ C2×C4 ▷ C2 ▷ C1 | |
601 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)(3,13)(4,18,20,44)(5,23)(6,11)(7,12)(8,15,26,37)(9,22,29,46)(10,31)(14,34)(16,42)(17,36)(19,32)(21,48,47,30)(24,33)(25,51)(27,54,52,63)(28,41,53,60)(35,55)(38,61)(39,58)(40,62,59,50)(43,57)(45,49,64,56);(1,3,5,16,6,17,24,43,7,14,25,38,19,39,31,55)(2,8,10,27,32,53,51,62,12,26,33,52,11,28,23,50)(4,15,21,40,22,41,49,63,20,37,47,59,46,60,56,54)(9,13,30,35,44,58,64,61,29,34,48,57,18,36,45,42);(3,14)(8,26)(13,34)(15,37)(16,38)(17,39)(27,52)(28,53)(35,57)(36,58)(40,59)(41,60)(42,61)(43,55)(50,62)(54,63)Polycyclic presentation:
2^9|a^2=e,b^2=f,d^2=ghi,f^2=gh,g^2=i,b^a=bd,d^a=dg,f^a=fghi,h^a=hi,c^b=ci,e^b=eh,e^d=ei68 conjugacy classes — view the full character table.
| maximal subgroup of | #72280870, #72280891, #72280958, … |
| maximal normal subgroup of | #72280870, #72280891, #72280958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 34538);