Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11836 |
| automorphism | Aut(G) = #72305983 (order 2048) · Inn(G) = #1514 · Out(G) = C24 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1514 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C2×M16 | C23 |
| prime radical | C23 | AGL(1,ℤ/8) |
| LCS | G ▷ C2×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C2×M16 ▷ C2×C4 ▷ C2 ▷ C1 | |
140 conjugacy classes of subgroups; only the 45 normal subgroups are shown.
This group has 140 conjugacy classes of subgroups — too many to draw. Only its 45 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12)(4,16,18,26)(5,8)(6,11)(7,13,23,31)(9,20,27,30)(10,19)(14,28)(15,29)(17,22,32,24)(21,25);(1,3,6,15)(2,7,11,23)(4,17,18,32)(5,21,19,14)(8,22,10,24)(9,25,27,28)(12,16,29,26)(13,30,31,20);(2,8,11,10)(4,18)(7,22,23,24)(9,26,27,16)(12,28,29,25)(13,31)(17,32)(20,30)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=fgh,d^2=gh,f^2=h,g^2=h,b^a=bd,c^a=cg,d^a=df,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,g^b=gh,d^c=dh,e^d=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #139424, #139425, #139428, … |
| maximal normal subgroup of | #139424, #139425, #139428, … |
| maximal quotient of | #124542, #124635, #124642, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4826);