Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #12310 |
| automorphism | Aut(G) = #72309267 (order 4096) · Inn(G) = #1619 · Out(G) = C2×C22⋊C4 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1619 |
| commutator, Frattini, cosocle | C4×C8 | C23 |
| prime radical | #3293 | C2 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
178 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,7,3)(4,14,17,30)(5,19,24,20)(6,13,22,10)(8,25,11,16)(9,28,12,23)(15,29,18,31)(21,26,32,27);(1,3,7,2)(4,11,17,8)(5,20,24,19)(6,10,22,13)(9,23,12,28)(14,21,30,32)(15,26,18,27)(16,31,25,29);(1,4)(2,8)(3,11)(5,21)(6,15)(7,17)(9,29)(10,26)(12,31)(13,27)(14,19)(16,23)(18,22)(20,30)(24,32)(25,28)Polycyclic presentation:
2^8|a^2=h,b^2=h,d^2=f,e^2=gh,g^2=h,b^a=bd,c^a=ce,d^a=df,e^a=eg,g^a=gh,d^b=df,e^b=eg,g^b=gh,d^c=dg,e^c=eg,f^c=fh,g^c=gh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94822, #153620, #153622, … |
| maximal normal subgroup of | #94822, #153620, #153622, … |
| maximal quotient of | #123303, #123304, #123305, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5301);