Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
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 | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
477 conjugacy classes of subgroups; only the 183 normal subgroups are shown.
This group has 477 conjugacy classes of subgroups — too many to draw. Only its 183 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)(5,6,8,7)(9,10)(11,18)(12,16)(13,22)(14,17)(15,19)(20,23)(21,24);(9,11,13,19,14,20,12,21)(10,15,16,18,17,24,22,23);(2,4)(9,10)(11,15)(12,22)(13,16)(14,17)(18,19)(20,24)(21,23);(5,7,8,6)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,24,18)Polycyclic presentation:
2^8|a^2=g,b^2=e,d^2=gh,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 | #134476, #134506, #495724, … |
| maximal normal subgroup of | #134476, #134506, #495724, … |
| maximal quotient of | #210732, #210738, #210756, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14518);