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 | C24 | C22⋊C4 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4⋊C4 ▷ C23 ▷ C2 ▷ C1 | |
183 conjugacy classes of subgroups; only the 49 normal subgroups are shown.
This group has 183 conjugacy classes of subgroups — too many to draw. Only its 49 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:
(1,2,4,7,5,8,11,14)(3,6,10,13,9,12,15,16)(17,18)(19,27)(20,24)(21,25,30,31)(22,26)(23,28,32,29);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,22,20)(18,23,26,32)(21,29,30,28)(24,31,27,25);(3,9)(6,12)(10,15)(13,16)(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=h,c^2=h,d^2=h,e^2=g,b^a=bd,d^a=df,f^a=fh,c^b=cg,d^b=dh,e^b=efh,e^d=eh37 conjugacy classes — view the full character table.
| maximal subgroup of | #132878, #132880, #132882, … |
| maximal normal subgroup of | #132878, #132880, #132882, … |
| maximal quotient of | #125071, #125087, #125104, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4540);