Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17547 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | #3497 | C2 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
601 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 601 conjugacy classes of subgroups — too many to draw. Only its 223 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) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,7)(6,8)(10,15,17,21)(16,22,24,23);(2,5)(3,8)(6,7)(9,10)(11,16)(12,15)(13,21)(14,17)(18,23)(19,24)(20,22);(9,11,14,19)(10,16,17,24)(12,20,13,18)(15,22,21,23);(9,12,14,13)(10,15,17,21)(11,18,19,20)(16,23,24,22)Polycyclic presentation:
2^8|a^2=f,c^2=gh,d^2=gh,e^2=h,b^a=be,e^a=eg,e^b=eh,f^b=fgh,d^c=dgh61 conjugacy classes — view the full character table.
| Fitting subgroup of | #17618⋊C3 |
| maximal subgroup of | #221145, #221146, #221149, … |
| maximal normal subgroup of | #221145, #221146, #221149, … |
| maximal quotient of | #160506, #160507, #161092, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10606);