Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2184 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C2×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| socle | C22 | C22×D8 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
298 conjugacy classes of subgroups; only the 105 normal subgroups are shown.
This group has 298 conjugacy classes of subgroups — too many to draw. Only its 105 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 — 4 generators on 24 points:
(1,2)(3,6)(4,5)(7,8)(10,14)(11,17)(13,20)(15,23)(16,19)(18,24)(21,22);(2,5)(6,8)(9,10,12,14)(11,15,18,22)(13,16,19,20)(17,21,24,23);(9,11)(10,15)(12,18)(13,21)(14,22)(16,24)(17,20)(19,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|b^2=h,d^2=eh,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,f^c=fg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #137009, #137010, #137017, … |
| maximal normal subgroup of | #137009, #137010, #137017, … |
| maximal quotient of | #494028, #494964, #494966, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26654);