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)(3,8,14,23)(4,16)(5,12)(6,11)(7,10)(9,18)(13,27,29,31)(15,25,19,22)(17,26)(20,24)(21,30,32,28);(1,3,6,15,7,14,5,19)(2,8,12,22,10,23,11,25)(4,13,18,30,17,29,20,28)(9,21,16,31,24,32,26,27);(1,4)(2,9)(3,13)(5,20)(6,18)(7,17)(8,21)(10,24)(11,26)(12,16)(14,29)(15,30)(19,28)(22,31)(23,32)(25,27)Polycyclic presentation:
2^8|a^2=f,b^2=g,e^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 | #129779, #129780, #137966, … |
| maximal normal subgroup of | #129779, #129780, #137966, … |
| maximal quotient of | #105682, #105683, #105686, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3781);