Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | C2×#383 |
| prime radical | C24 | #41 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
342 conjugacy classes of subgroups; only the 181 normal subgroups are shown.
This group has 342 conjugacy classes of subgroups — too many to draw. Only its 181 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,12)(4,16,18,23)(5,10)(6,11)(7,15)(8,17,24,30)(9,19)(13,20,26,31)(14,22)(21,28)(25,27,29,32);(1,3,6,14)(2,7)(4,13,18,26)(5,15,19,28)(8,20)(9,12)(10,22)(11,21)(16,29)(17,27,30,32)(23,25)(24,31);(1,4,5,17,6,18,19,30)(2,8,9,23,11,24,10,16)(3,13,15,27,14,26,28,32)(7,20,12,25,21,31,22,29);(3,14)(7,21)(12,22)(13,26)(15,28)(20,31)(25,29)(27,32)Polycyclic presentation:
2^8|a^2=g,b^2=ef,c^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #138390, #138391, #361226, … |
| maximal normal subgroup of | #138390, #138391, #361226, … |
| maximal quotient of | #191287, #191288, #191291, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13764);