Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 65536 |
| 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×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
719 conjugacy classes of subgroups; only the 271 normal subgroups are shown.
This group has 719 conjugacy classes of subgroups — too many to draw. Only its 271 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)(12,19)(15,20)(16,21)(18,23)(22,24);(9,10,12,14,13,16,19,21)(11,15,18,20,17,22,23,24);(2,4)(9,11)(10,15)(12,18)(13,17)(14,20)(16,22)(19,23)(21,24);(10,16)(11,17)(14,21)(18,23)Polycyclic presentation:
2^8|a^2=g,b^2=e,e^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #138346, #138347, #361200, … |
| maximal normal subgroup of | #138346, #138347, #361200, … |
| maximal quotient of | #188712, #188719, #188726, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13652);