Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | Aut(G) = #72307301 (order 2048) · Inn(G) = #427 · Out(G) = C23×C4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #427 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| socle | C2 | #1568 |
| prime radical | C23 | #108 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ #97 ◁ #405 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 64 points:
(1,2,6,13,8,14,31,43,9,15,30,12,7,34,5,29)(3,16,19,36,21,45,18,37,22,41,32,58,20,54,52,10)(4,23,25,56,27,40,24,42,28,57,33,59,26,11,38,39)(17,46,48,35,50,60,62,55,51,64,53,61,49,44,47,63);(1,3,5,18,7,20,30,19,9,22,31,52,8,21,6,32)(2,10,12,16,14,36,29,45,15,37,13,41,34,58,43,54)(4,17,24,47,26,49,25,53,28,51,38,62,27,50,33,48)(11,35,39,44,23,55,56,46,40,61,42,60,57,63,59,64);(1,4,7,26,9,28,8,27)(2,11,14,23,15,40,34,57)(3,17,20,49,22,51,21,50)(5,24,30,25,31,38,6,33)(10,35,36,55,37,61,58,63)(12,39,29,56,13,42,43,59)(16,44,45,46,41,60,54,64)(18,47,19,53,52,62,32,48)Polycyclic presentation:
2^8|a^2=e,b^2=d,c^2=f,d^2=f,e^2=g,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,e^b=eh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94210, #94211, #134863, … |
| maximal normal subgroup of | #94210, #94211, #134863, … |
| maximal quotient of | #126700, #129106, #129115, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5014);