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 |
| LCS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
183 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(3,11)(4,15)(5,18)(7,21)(8,16)(9,20)(12,17)(13,29)(14,28)(19,25)(22,26)(23,32)(24,31)(27,30);(1,2,6,10)(3,7,13,23)(4,16,19,30)(5,20,18,9)(8,25,27,15)(11,26,29,31)(12,24,14,22)(17,21,28,32);(1,3)(2,7)(4,17)(5,12)(6,13)(8,26)(9,22)(10,23)(11,15)(14,18)(16,21)(19,28)(20,24)(25,29)(27,31)(30,32)Polycyclic presentation:
2^8|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 | #94821, #153619, #153621, … |
| maximal normal subgroup of | #94821, #153619, #153621, … |
| maximal quotient of | #123281, #123282, #123283, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5299);