Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3198 |
| 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, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
367 conjugacy classes of subgroups; only the 159 normal subgroups are shown.
This group has 367 conjugacy classes of subgroups — too many to draw. Only its 159 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)(13,14)(15,22)(16,20)(17,26)(18,21)(19,24)(23,27)(25,28);(1,3,4,2)(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,26)(17,20)(18,21)(22,24)(23,28)(25,27);(6,9)(10,12)(13,15,17,24,18,25,16,23)(14,19,20,22,21,27,26,28);(5,7,8,11)(6,10,9,12)(13,16,18,17)(14,20,21,26)(15,23,25,24)(19,22,27,28)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=f,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,d^b=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #219600, #219602, #219614, … |
| maximal normal subgroup of | #219600, #219602, #219614, … |
| maximal quotient of | #219080, #219087, #219112, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15217);