Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11663 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#102 |
| commutator | C2×C8 | C22×C4 |
| Frattini, cosocle | C4×C8 | C23 |
| prime radical | C23 | #102 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
181 conjugacy classes of subgroups; only the 65 normal subgroups are shown.
This group has 181 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2,3,6,4,5,7,8)(10,14,15,22)(11,17)(12,20)(16,18,21,23)(19,24);(9,10,13,15)(11,18,19,23)(12,21,20,16)(14,17,22,24);(2,5)(6,8)Polycyclic presentation:
2^8|a^2=e,b^2=h,d^2=f,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=df,f^a=fh,d^b=dfh,e^b=eh,f^b=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131043, #131049, #131067, … |
| maximal normal subgroup of | #131043, #131049, #131067, … |
| maximal quotient of | #124663, #124701, #124733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4652);