Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11247 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #387 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1443 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
274 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 274 conjugacy classes of subgroups — too many to draw. Only its 101 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 — 3 generators on 32 points:
(1,2)(3,11,14,7)(4,8,16,23)(5,10)(6,9)(12,27)(13,28,31,21)(15,32)(17,24,18,26)(19,25)(20,29)(22,30);(1,3)(2,7)(4,15,16,30)(5,13)(6,14)(8,22,23,32)(9,11)(10,21)(12,17,29,18)(19,31)(20,24,27,26)(25,28);(1,4)(2,8,9,23)(3,12)(5,18)(6,16)(7,20,11,27)(10,26,25,24)(13,30)(14,29)(15,31)(17,19)(21,32,28,22)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,c^b=cf,g^b=gh,e^c=eh,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11309, #358643, #360368, … |
| maximal normal subgroup of | C2×#11309, #358643, #360368, … |
| maximal quotient of | #121081, #121099, #121101, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4297);