Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3057 |
| automorphism | Aut(G) order 16384 |
| 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 | |
307 conjugacy classes of subgroups; only the 161 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 161 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,3,4,6,8,7)(9,10)(11,19)(12,16)(13,25)(14,17)(15,21)(18,27)(20,33)(22,32)(23,31)(24,37)(26,29)(28,35)(30,36)(34,40)(38,39);(2,6)(3,7)(9,11,14,22)(10,15,17,29)(12,20,23,34)(13,26,27,21)(16,28,31,39)(18,32,25,19)(24,38,36,35)(30,40,37,33);(1,3,5,6,4,7,8,2)(9,12,14,23)(10,16,17,31)(11,20,22,34)(13,24,27,36)(15,28,29,39)(18,30,25,37)(19,33,32,40)(21,35,26,38);(10,17)(12,23)(15,29)(18,25)(19,32)(20,34)(24,36)(35,38)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=fg,e^2=g,f^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,c^b=ch,e^b=eg,d^c=dg64 conjugacy classes — view the full character table.
| maximal subgroup of | #135038, #135041, #135071, … |
| maximal normal subgroup of | #135038, #135041, #135071, … |
| maximal quotient of | #182422, #183624, #183933, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10885);