Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3055 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
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×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
417 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 417 conjugacy classes of subgroups — too many to draw. Only its 171 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,23)(16,19)(17,27)(18,22)(20,25)(21,24)(26,28);(6,9)(10,12)(13,15,24,26,18,25,17,16)(14,19,27,20,22,28,21,23);(5,7,8,11)(6,10,9,12)(13,16,17,25,18,26,24,15)(14,20,21,19,22,23,27,28);(1,3,4,2)(15,25)(16,26)(19,28)(20,23)Polycyclic presentation:
2^8|a^2=g,b^2=fh,c^2=e,d^2=g,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,d^b=dh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #149706, #149866, #151940, … |
| maximal normal subgroup of | #149706, #149866, #151940, … |
| maximal quotient of | #212593, #212599, #212605, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14636);