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 | #3292 | 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 | |
186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,6,10)(3,11,14,30)(4,16,20,32)(5,21,19,9)(7,22,24,13)(8,26,28,17)(12,27,15,29)(18,23,31,25);(4,17)(5,19)(8,16)(9,21)(11,27)(12,15)(13,18)(20,26)(22,31)(23,25)(28,32)(29,30);(1,3)(2,7)(4,18)(5,12)(6,14)(8,27)(9,23)(10,24)(11,16)(13,17)(15,19)(20,31)(21,25)(22,26)(28,29)(30,32)Polycyclic presentation:
2^8|a^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 | #94820, #153614, #153616, … |
| maximal normal subgroup of | #94820, #153614, #153616, … |
| maximal quotient of | #123295, #123296, #123297, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5300);