Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 240 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) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 24 points:
(1,2)(3,6,8,5)(4,7)(9,10,13,17,14,18,12,16)(11,15,19,22,20,23,21,24);(2,5,7,6)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,22)(18,23);(9,12,14,13)(10,16,18,17)(11,19,20,21)(15,22,23,24)Polycyclic presentation:
2^8|a^2=e,b^2=fh,c^2=g,d^2=h,e^2=g,b^a=bd,d^a=df,f^a=fh,c^b=cg,e^b=efh,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132358, #132360, #132362, … |
| maximal normal subgroup of | #132358, #132360, #132362, … |
| maximal quotient of | #125188, #125218, #125239, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4558);