Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3192 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
363 conjugacy classes of subgroups; only the 163 normal subgroups are shown.
This group has 363 conjugacy classes of subgroups — too many to draw. Only its 163 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) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2,4,3)(6,9)(7,11)(10,12)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,28)(25,27);(1,3,4,2)(13,15)(14,19)(16,25)(17,23)(18,24)(20,28)(21,22)(26,27);(5,6,7,9,8,10,11,12)(13,16,18,17)(14,20,21,26)(15,23,24,25)(19,27,22,28);(6,10)(9,12)(13,17,18,16)(14,20,21,26)(15,23,24,25)(19,28,22,27)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #100048, #100056, #209153, … |
| maximal normal subgroup of | #100048, #100056, #209153, … |
| maximal quotient of | #219256, #219262, #219277, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15238);