Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C22⋊C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
346 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 346 conjugacy classes of subgroups — too many to draw. Only its 161 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 — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,14,16,20)(11,17)(15,22,23,18)(19,24);(1,3,4,7)(2,5,6,8)(9,10)(11,18)(12,20)(13,16)(14,21)(15,19)(17,22)(23,24);(2,6)(5,8);(9,11,12,19,13,17,21,24)(10,15,14,18,16,23,20,22)Polycyclic presentation:
2^8|a^2=g,b^2=f,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=dh,d^b=de,e^b=eh,g^b=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #138974, #138975, #138986, … |
| maximal normal subgroup of | #138974, #138975, #138986, … |
| maximal quotient of | #225232, #225233, #225234, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15595);