Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #7225 |
| automorphism | Aut(G) = C4×#23882 (order 1024) · Inn(G) = H(3,ℤ/4) · Out(G) = C22×C4 |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | H(3,ℤ/4) |
| commutator | C2×C8 | C42 |
| Frattini, cosocle | C4×M16 | C22 |
| socle | C2 | #1288 |
| prime radical | C23 | #104 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×C8 ◁ G | |
| LECS | G ▷ C4×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
90 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,11,13,24)(4,15,16,29)(5,9)(6,10)(7,12)(8,17,22,30)(14,25,28,32)(18,21)(19,26)(20,27)(23,31);(1,3)(2,7,9,19,10,20,21,23)(4,12)(5,14)(6,13)(8,11,15,25,22,24,29,32)(16,27)(17,26)(18,28)(30,31)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fgh,e^2=gh,f^2=h,b^a=bc,c^a=cf,e^a=efgh,g^a=gh,d^b=dg,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#7226, #127314, #127315, … |
| maximal normal subgroup of | C2×#7226, #127314, #127315, … |
| maximal quotient of | #93318, #93354, #93360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 214);