Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| socle | C22 | C2×#99 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
201 conjugacy classes of subgroups; only the 53 normal subgroups are shown.
This group has 201 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:
(2,6,7,8)(9,10,13,17,14,16,21,23)(11,18,19,24,12,15,20,22);(1,2)(3,6)(4,8)(5,7)(9,11,14,12)(10,15,16,18)(13,19,21,20)(17,24,23,22);(1,3,5,4)(2,6,7,8)(9,12,14,11)(10,15,16,18)(13,20,21,19)(17,22,23,24)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=gh,d^2=g,e^2=h,b^a=bd,d^a=df,f^a=fh,d^b=dg,e^b=efg,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #128968, #128969, #128980, … |
| maximal normal subgroup of | #128968, #128969, #128980, … |
| maximal quotient of | #124140, #124150, #124170, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4724);