Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1812 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C8 | #117 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C2 | #2283 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C8 ◁ #405 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
308 conjugacy classes of subgroups; only the 157 normal subgroups are shown.
This group has 308 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)(3,12,15,8)(4,9)(5,10)(6,11)(7,14,20,28)(13,25,27,21)(16,22)(17,23)(18,24)(19,26,30,32)(29,31);(1,3)(2,7,11,20)(4,13)(5,14)(6,15)(8,10,12,24)(9,19,23,30)(16,26)(17,27)(18,28)(21,22,25,31)(29,32);(1,3)(2,8)(4,13)(5,14)(6,15)(7,10)(9,21)(11,12)(16,26)(17,27)(18,28)(19,22)(20,24)(23,25)(29,32)(30,31);(1,4,5,16,6,17,18,29)(2,9,10,22,11,23,24,31)(3,13,14,26,15,27,28,32)(7,19,12,25,20,30,8,21)Polycyclic presentation:
2^8|a^2=g,b^2=f,d^2=e,e^2=h,b^a=be,c^a=cf,f^a=fh,g^b=gh,g^c=gh64 conjugacy classes — view the full character table.
| maximal subgroup of | #126850, #126851, #126872, … |
| maximal normal subgroup of | #126850, #126851, #126872, … |
| maximal quotient of | #191325, #191326, #191327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13775);