Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24×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 | C2×#121 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
575 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 575 conjugacy classes of subgroups — too many to draw. Only its 199 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:
(1,2,3,4)(6,9)(7,11)(13,14)(15,22)(16,20)(17,19)(18,23)(21,25)(24,26)(27,28);(5,6)(7,10)(8,9)(11,12)(13,15)(14,18)(16,23)(17,24)(19,27)(20,22)(21,26)(25,28);(6,9)(10,12)(14,19)(15,23)(17,25)(18,26)(20,21)(22,27);(5,7,8,11)(6,10,9,12)(14,20)(18,22)(19,21)(26,27)Polycyclic presentation:
2^8|a^2=g,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #107873, #108357, #119563, … |
| maximal normal subgroup of | #107873, #108357, #119563, … |
| maximal quotient of | #216330, #216336, #216360, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14874);