Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10530 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #394 |
| commutator, Frattini, cosocle | C2×C42 | C23 |
| prime radical | #3253 | C2 |
| LCS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
201 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 201 conjugacy classes of subgroups — too many to draw. Only its 49 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 — 3 generators on 24 points:
(1,2)(3,6)(4,7)(5,8)(9,10)(11,21)(12,16)(13,19)(14,18)(15,20)(17,23)(22,24);(1,2)(3,7)(4,6)(5,8)(9,11)(10,16)(12,14)(13,23)(15,22)(17,20)(18,24)(19,21);(2,6)(3,4)(7,8)(9,12,13,22,15,23,14,11)(10,17,18,24,20,16,19,21)Polycyclic presentation:
2^8|c^2=d,d^2=g,e^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^a=fg,c^b=cf,d^b=dg,e^b=eg,f^b=fgh,e^c=egh,f^c=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #94686, #94688, #151278, … |
| maximal normal subgroup of | #94686, #94688, #151278, … |
| maximal quotient of | #104877, #104884, #104895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3518);