Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11992 |
| automorphism | Aut(G) = #72307074 (order 4096) · Inn(G) = #1555 · Out(G) = C25 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #1555 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | #532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
180 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 180 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,4,10,5,9,18,25,6,8,17,24,20,11,28,27)(3,12,13,29,14,21,31,32,15,23,30,26,19,7,16,22);(1,3,6,15)(2,7,8,21)(4,16,17,31)(5,19,20,14)(9,23,11,12)(10,26,24,29)(13,18,30,28)(22,25,32,27);(2,8)(4,17)(7,21)(9,11)(12,23)(13,30)(16,31)(18,28)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=g,e^2=fg,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,e^c=eh28 conjugacy classes — view the full character table.
| maximal subgroup of | #94787, #94789, #94790, … |
| maximal normal subgroup of | #94787, #94789, #94790, … |
| maximal quotient of | #124522, #124523, #124630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4981);