Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10792 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C4×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C2 | C22⋊#95 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
582 conjugacy classes of subgroups; only the 173 normal subgroups are shown.
This group has 582 conjugacy classes of subgroups — too many to draw. Only its 173 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:
(1,2,5,12,7,10,6,11)(3,8,15,23,14,24,20,22)(4,16,19,25,17,26,18,9)(13,27,29,32,30,31,28,21);(1,3,6,15,7,14,5,20)(2,8,12,22,10,24,11,23)(4,13,18,29,17,30,19,28)(9,21,16,31,25,32,26,27);(1,4)(2,9)(3,13)(5,19)(6,18)(7,17)(8,21)(10,25)(11,26)(12,16)(14,30)(15,29)(20,28)(22,31)(23,27)(24,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=eh58 conjugacy classes — view the full character table.
| maximal subgroup of | #129777, #129778, #137964, … |
| maximal normal subgroup of | #129777, #129778, #137964, … |
| maximal quotient of | #105682, #105683, #105684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3782);