Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1304 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #95 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | C2×#95 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C22 ▷ C1 | |
281 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 281 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2,5,10,6,9,4,8)(3,11,12,16,13,7,14,15)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,30)(29,32);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19)(18,22,24,25)(20,26)(21,27)(23,29,30,31)(28,32);(1,4,6,5)(2,8,9,10)(3,12,13,14)(7,15,11,16)(17,20,21,28)(18,23,24,30)(19,26,27,32)(22,29,25,31)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=gh,e^2=g,b^a=bd,d^a=dg,f^a=fh,d^b=dh,e^b=eg52 conjugacy classes — view the full character table.
| maximal subgroup of | #105476, #105477, #105847, … |
| maximal normal subgroup of | #105476, #105477, #105847, … |
| maximal quotient of | #95107, #95111, #95125, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1094);