Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33758 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
286 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 286 conjugacy classes of subgroups — too many to draw. Only its 101 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:
(2,5)(3,7)(9,10)(11,15)(12,16)(13,22)(14,18)(17,20)(19,23)(21,24);(1,2)(3,6)(4,5)(7,8)(9,11,14,13)(10,15,18,22)(12,19,21,20)(16,23,24,17);(2,5)(6,8)(10,16)(11,13)(12,20)(15,17)(18,24)(19,21)(22,23);(1,3,4,7)(2,6,5,8)Polycyclic presentation:
2^8|b^2=h,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=de,f^a=fg,g^a=gh,c^b=ceh,f^c=fg,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #143606, #143610, #143614, … |
| maximal normal subgroup of | #143606, #143610, #143614, … |
| maximal quotient of | #496593, #496702, #496910, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26807);