Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20671 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | C2×#117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C2×C8 | C24 |
| socle | C2 | C2×#432 |
| prime radical | C2×#41 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C2×#97 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
260 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 260 conjugacy classes of subgroups — too many to draw. Only its 157 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 — 4 generators on 32 points:
(1,2,4,10,5,11,18,27,6,9,17,25,19,8,23,24)(3,12,14,26,15,20,29,31,13,7,16,21,28,22,30,32);(1,3)(2,7)(4,16)(5,15)(6,13)(8,20)(9,12)(10,26)(11,22)(14,17)(18,30)(19,28)(21,25)(23,29)(24,32)(27,31);(2,8,9,11)(4,17)(7,20,12,22)(10,27,25,24)(14,16)(18,23)(21,32,26,31)(29,30);(3,13)(7,12)(14,16)(15,28)(20,22)(21,26)(29,30)(31,32)Polycyclic presentation:
2^8|a^2=f,c^2=e,f^2=g,g^2=h,b^a=be,c^a=cg,e^a=eh,d^b=dh,f^b=fh,f^c=fh58 conjugacy classes — view the full character table.
| maximal subgroup of | #138092, #138093, #480440, … |
| maximal normal subgroup of | #138092, #138093, #480440, … |
| maximal quotient of | #492928, #492955, #493109, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26414);