Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23414 |
| automorphism | Aut(G) = C22×#72224760 (order 6144) · Inn(G) = C2×#127 · Out(G) = GL(2,ℤ/4) |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#127 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2397 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
454 conjugacy classes of subgroups; only the 125 normal subgroups are shown.
This group has 454 conjugacy classes of subgroups — too many to draw. Only its 125 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:
(1,2,5,9,6,10,18,24)(3,11,13,19,14,23,17,7)(4,8,16,22,15,21,28,31)(12,25,27,30,26,32,29,20);(1,3)(2,7)(4,12)(5,17)(6,14)(8,20)(9,23)(10,19)(11,24)(13,18)(15,26)(16,29)(21,30)(22,32)(25,31)(27,28);(4,15)(7,19)(8,21)(11,23)(12,26)(16,28)(22,31)(27,29);(1,4,5,16,6,15,18,28)(2,8,9,22,10,21,24,31)(3,12,17,29,14,26,13,27)(7,20,23,32,19,30,11,25)Polycyclic presentation:
2^8|a^2=e,d^2=ef,e^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,e^b=eh,f^b=fh,d^c=dh46 conjugacy classes — view the full character table.
| Fitting subgroup of | #23795⋊C3 |
| maximal subgroup of | #486082, #486083, C2×#23795, … |
| maximal normal subgroup of | #486082, #486083, C2×#23795, … |
| maximal quotient of | #244141, #244142, #244143, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16783);