Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C42 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C23 | C4×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ #431 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
293 conjugacy classes of subgroups; only the 149 normal subgroups are shown.
This group has 293 conjugacy classes of subgroups — too many to draw. Only its 149 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) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,4,8,5,7,11,14)(3,6,9,13,10,12,15,16)(17,18,22,25,23,19,21,27)(20,24,30,31,28,26,29,32);(17,19,23,18)(20,26,28,24)(21,25,22,27)(29,31,30,32);(1,3,4,9,5,10,11,15)(2,6,8,13,7,12,14,16)(17,20,21,29,23,28,22,30)(18,24,27,32,19,26,25,31);(2,7)(6,12)(8,14)(13,16)(20,28)(24,26)(29,30)(31,32)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fg,e^2=g,f^2=h,b^a=be,d^a=dgh,e^a=eg,e^b=eg,d^c=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17925, #6344637, #6344645, … |
| maximal normal subgroup of | C2×#17925, #6344637, #6344645, … |
| maximal quotient of | #182506, #182542, #183630, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10913);