Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #9755 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #390 |
| commutator | C42 | C22×C4 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C23 | #124 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
199 conjugacy classes of subgroups; only the 71 normal subgroups are shown.
This group has 199 conjugacy classes of subgroups — too many to draw. Only its 71 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) = 128 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,9)(4,13)(5,7)(6,11)(8,10)(12,15)(14,16)(17,18,21,25,23,26,22,32)(19,20,27,29,28,30,31,24);(1,3,10,16,5,12,4,11)(2,6,13,15,7,14,8,9)(17,19,23,28)(18,24,26,29)(20,25,30,32)(21,31,22,27);(2,7)(6,14)(8,13)(9,15)(17,20,22,24,23,30,21,29)(18,19,25,27,26,28,32,31)Polycyclic presentation:
2^8|a^2=f,b^2=dfg,c^2=eg,d^2=gh,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,f^b=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#9770, #205641, #205644, … |
| maximal normal subgroup of | C2×#9770, #205641, #205644, … |
| maximal quotient of | #102034, #102047, #102214, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2758);