Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #11549 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C2×C4⋊C4 | C23 |
| prime radical | C23 | C2×C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
175 conjugacy classes of subgroups; only the 51 normal subgroups are shown.
This group has 175 conjugacy classes of subgroups — too many to draw. Only its 51 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) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,9,6,10,4,8)(3,7,11,14,12,15,13,16)(17,18)(19,27)(20,24)(21,25,30,31)(22,26)(23,28,32,29);(1,3)(2,7)(4,13)(5,11)(6,12)(8,16)(9,14)(10,15)(17,19,22,20)(18,23)(21,28,30,29)(24,31)(25,27)(26,32);(1,4,6,5)(2,8,10,9)(3,11,12,13)(7,14,15,16)(17,20,22,19)(18,24,26,27)(21,29,30,28)(23,31,32,25)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=gh,d^2=h,e^2=g,b^a=bd,d^a=df,f^a=fh,c^b=cg,e^b=efh,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132417, #132418, #132419, … |
| maximal normal subgroup of | #132417, #132418, #132419, … |
| maximal quotient of | #125189, #125219, #125238, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4559);