Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) = C25×C4×D4 (order 1024) · Inn(G) = C2×D4 · Out(G) = C24×C4 |
| homology | Abelianization (H1) = C22×C8 · Schur multiplier (H2) = C2 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C2×D4 |
| commutator | C2×C4 | C22×C8 |
| Frattini, cosocle | C4×C8 | C23 |
| socle, prime radical | C22 | C8×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C4×C16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
102 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18,20,24,21,19,26,27,22,25,29,32,30,28,23,31);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9);(2,7)(6,15)(8,12)(9,16)(17,19,20,27,21,25,26,32,22,28,29,31,30,18,23,24)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,d^2=f,e^2=g,g^2=h,b^a=bd,c^a=cfh,d^a=df,d^b=df76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#12060, #483571, #483572, … |
| maximal normal subgroup of | C2×#12060, #483571, #483572, … |
| maximal quotient of | #128697, #128699, #129163, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5048);