Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2081 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #397 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1442 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
288 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 288 conjugacy classes of subgroups — too many to draw. Only its 109 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:
(1,2,6,10,7,11,5,20)(3,12,14,8,15,24,21,23)(4,16,19,31,18,9,17,25)(13,27,28,30,29,26,32,22);(1,3)(2,8)(4,13,18,29)(5,21)(6,14)(7,15)(9,22,16,30)(10,24)(11,23)(12,20)(17,32,19,28)(25,27,31,26);(1,4,5,17,7,18,6,19)(2,9,10,25,11,16,20,31)(3,13,14,28,15,29,21,32)(8,22,12,26,23,30,24,27)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^b=fh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11284, #359281, #360824, … |
| maximal normal subgroup of | C2×#11284, #359281, #360824, … |
| maximal quotient of | #120675, #120676, #120735, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4272);