Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 5 |
| isoclinism | isoclinic class of #1412 |
| automorphism | Aut(G) = #72318745 (order 4096) · Inn(G) = #354 · Out(G) = #534 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #354 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #127 | C23 |
| socle | C2 | C2×#354 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ G | |
| LECS | G ▷ #127 ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
163 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,15,22)(4,8)(5,10,19,9)(6,11)(7,14,21,18)(13,26,28,32)(16,24,30,23)(17,25)(20,27,31,29);(1,3,6,14)(2,7)(4,13,17,27)(5,18,19,15)(8,20)(9,12)(10,21)(11,22)(16,29,30,28)(23,26)(24,31)(25,32);(1,4,6,17)(2,8,11,25)(3,13,14,27)(5,16,19,30)(7,20,22,32)(9,23,10,24)(12,26,21,31)(15,28,18,29)Polycyclic presentation:
2^8|a^2=e,b^2=g,c^2=h,d^2=gh,f^2=h,b^a=bd,d^a=df,f^a=fg,g^a=gh,d^b=dg,e^b=efg,f^b=fh,e^d=egh,f^e=fh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94784, #151937, #151938, … |
| maximal normal subgroup of | #94784, #151937, #151938, … |
| maximal quotient of | #149582, #149614, #149702, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 6582);