Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11115 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #388 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1440 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
402 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 402 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:
(1,2,6,10,7,12,5,11)(3,13,15,8,17,24,16,25)(4,18,20,28,21,27,22,9)(14,30,31,26,32,23,19,29);(1,3)(2,8)(4,19)(5,16)(6,15)(7,17)(9,26)(10,24)(11,13)(12,25)(14,20)(18,23)(21,31)(22,32)(27,30)(28,29);(1,4,5,20,7,21,6,22)(2,9,10,27,12,28,11,18)(3,14,15,19,17,32,16,31)(8,23,13,29,25,30,24,26)Polycyclic presentation:
2^8|a^2=g,c^2=d,d^2=h,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,f^b=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11179, #358422, #359902, … |
| maximal normal subgroup of | C2×#11179, #358422, #359902, … |
| maximal quotient of | #118683, #118760, #118761, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4167);