Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2031 |
| automorphism | Aut(G) = C4×#511819 (order 2048) · Inn(G) = #124 · Out(G) = C2×C4×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #124 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C2 | #1832 |
| prime radical | C23 | #131 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,12,7,13,30,45,8,14,31,46,5,11,27,42)(3,15,19,47,17,36,51,43,20,34,28,59,18,9,32,35)(4,21,25,54,24,41,57,44,26,39,29,60,23,10,37,40)(16,48,49,62,22,55,56,64,50,33,53,61,52,38,58,63);(1,3,5,18,8,20,7,17)(2,9,11,34,14,36,13,15)(4,22,24,50,26,52,23,16)(6,28,27,51,31,19,30,32)(10,38,21,48,41,55,39,33)(12,43,42,47,46,35,45,59)(25,58,57,49,29,56,37,53)(40,64,54,61,44,63,60,62);(1,4,5,23,8,26,7,24)(2,10,11,39,14,41,13,21)(3,16,17,22,20,50,18,52)(6,29,27,57,31,25,30,37)(9,33,15,38,36,48,34,55)(12,44,42,54,46,40,45,60)(19,53,51,58,28,49,32,56)(35,62,47,64,43,61,59,63)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=d,d^2=h,e^2=h,f^2=g,g^2=h,b^a=bd,c^a=ce,c^b=cg,d^b=dh,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #145499, #145500, #145501, … |
| maximal normal subgroup of | #145499, #145500, #145501, … |
| maximal quotient of | #135640, #135641, #135642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5873);