Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #11000 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #390 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | #1439 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
418 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 418 conjugacy classes of subgroups — too many to draw. Only its 117 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)(3,12,16,23,15,28,14,8)(4,17,18,25,19,9,21,26)(13,27,29,22,30,32,31,24);(1,3,5,14,7,15,6,16)(2,8,10,12,11,23,20,28)(4,13,18,29,19,30,21,31)(9,24,25,32,17,22,26,27);(1,4)(2,9)(3,13)(5,18)(6,21)(7,19)(8,22)(10,25)(11,17)(12,27)(14,29)(15,30)(16,31)(20,26)(23,24)(28,32)Polycyclic presentation:
2^8|a^2=g,b^2=d,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,e^b=eh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #131352, #131353, #144694, … |
| maximal normal subgroup of | #131352, #131353, #144694, … |
| maximal quotient of | #116429, #116449, #116450, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4091);