Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C22 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #95 |
| commutator | C23 | C4×C8 |
| Frattini, cosocle | C23×C8 | C22 |
| socle | C22 | #336 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C23×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
157 conjugacy classes of subgroups; only the 55 normal subgroups are shown.
This group has 157 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 40 points:
(1,2)(3,7,8,5)(4,6)(9,10,12,16,13,18,23,30,14,17,24,29,25,31,36,39)(11,19,21,32,22,33,35,40,20,15,26,27,34,28,37,38);(1,3)(2,5)(4,8)(6,7)(9,11)(10,15,17,19)(12,21)(13,22)(14,20)(16,27,29,32)(18,28,31,33)(23,35)(24,26)(25,34)(30,38,39,40)(36,37)Polycyclic presentation:
2^8|a^2=d,b^2=e,d^2=g,g^2=h,b^a=bc,c^a=cf,e^a=eh,c^b=ch,d^b=df52 conjugacy classes — view the full character table.
| maximal subgroup of | #93902, #93907, #93942, … |
| maximal normal subgroup of | #93902, #93907, #93942, … |
| maximal quotient of | #92979, #92980, #92981, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 173);