Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | Aut(G) = #72306296 (order 2048) · Inn(G) = #329 · Out(G) = C22×D4 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #329 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | #97 | C23 |
| socle | C2 | C2×#329 |
| prime radical | #532 | C4 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ #97 ▷ C2×C4 ▷ C2 ▷ C1 | |
159 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 159 conjugacy classes of subgroups — too many to draw. Only its 53 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:
(1,2,5,10,4,8,17,25,6,11,20,29,18,9,26,27)(3,12,14,30,15,7,21,22,16,23,31,28,13,24,19,32);(1,3)(2,7)(4,13)(5,19)(6,16)(8,12)(9,23)(10,28)(11,24)(14,26)(15,18)(17,31)(20,21)(22,25)(27,32)(29,30);(1,4,6,18)(2,8,11,9)(3,13,16,15)(5,17,20,26)(7,12,24,23)(10,25,29,27)(14,19,31,21)(22,30,32,28)Polycyclic presentation:
2^8|a^2=e,c^2=h,d^2=g,e^2=f,f^2=h,b^a=bd,d^a=df,g^a=gh,d^b=dg,e^b=efgh,f^b=fh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94784, #94786, #128887, … |
| maximal normal subgroup of | #94784, #94786, #128887, … |
| maximal quotient of | #124062, #124118, #124205, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4899);