Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12026 |
| automorphism | Aut(G) = #72307299 (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:
(1,2,6,13,8,14,32,45,9,15,31,12,7,35,5,30)(3,16,19,37,21,47,18,38,22,43,33,61,20,55,53,10)(4,23,26,57,28,42,25,44,29,58,34,62,27,11,39,41)(17,48,50,36,52,63,60,64,24,59,54,40,51,46,49,56);(1,3,5,18,7,20,31,19,9,22,32,53,8,21,6,33)(2,10,12,16,14,37,30,47,15,38,13,43,35,61,45,55)(4,24,25,60,27,52,26,50,29,17,39,49,28,51,34,54)(11,40,41,63,23,56,57,59,42,36,44,46,58,64,62,48);(1,4,7,27,9,29,8,28)(2,11,14,23,15,42,35,58)(3,17,20,51,22,24,21,52)(5,25,31,26,32,39,6,34)(10,36,37,64,38,40,61,56)(12,41,30,57,13,44,45,62)(16,46,47,48,43,63,55,59)(18,49,19,54,53,60,33,50)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,c^b=ch,e^b=eh,e^c=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #94255, #94256, #134867, … |
| maximal normal subgroup of | #94255, #94256, #134867, … |
| maximal quotient of | #126699, #129107, #129116, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5015);