Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #468 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C42 | C2×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C2×#135 | C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C42 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
333 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 333 conjugacy classes of subgroups — too many to draw. Only its 163 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:
(1,2,5,3,4,6,8,7)(9,10)(11,18)(12,17)(13,21)(14,16)(15,19)(20,24)(22,23);(2,6)(3,7)(9,11)(10,15)(12,19)(13,22)(14,20)(16,23)(17,24)(18,21);(1,3,5,6,4,7,8,2)(10,16)(15,23)(17,21)(18,24);(9,12,14,13)(10,17,16,21)(11,19,20,22)(15,24,23,18)Polycyclic presentation:
2^8|a^2=f,c^2=f,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cgh,e^a=eg,c^b=ch,e^b=eg64 conjugacy classes — view the full character table.
| maximal subgroup of | #128740, #128741, #128755, … |
| maximal normal subgroup of | #128740, #128741, #128755, … |
| maximal quotient of | #182416, #183448, C4×#1592, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10878);