Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#135 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
733 conjugacy classes of subgroups; only the 207 normal subgroups are shown.
This group has 733 conjugacy classes of subgroups — too many to draw. Only its 207 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)(3,4)(5,6,7,8)(10,14)(12,20)(15,21)(16,17)(18,24)(22,23);(9,10,13,17)(11,15,19,23)(12,16,20,14)(18,22,24,21);(2,4)(9,11,13,19)(10,15,17,23)(12,18,20,24)(14,21,16,22);(10,16)(12,20)(14,17)(15,22)(18,24)(21,23)Polycyclic presentation:
2^8|a^2=g,b^2=h,c^2=h,e^2=h,b^a=be,c^a=cf,e^a=eh,d^b=de,e^b=eh,e^d=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #220529, #220531, #220549, … |
| maximal normal subgroup of | #220529, #220531, #220549, … |
| maximal quotient of | #220119, #220135, #220151, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15427);