Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2002 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #120 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #379 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
425 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 425 conjugacy classes of subgroups — too many to draw. Only its 127 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) = 64 |
A minimal faithful permutation representation — 3 generators on 40 points:
(9,10,14,18,15,19,13,26)(11,20,22,16,23,31,27,30)(12,17,24,33,25,34,28,32)(21,36,39,40,38,35,37,29);(1,2,4,6)(3,7,8,5)(9,11)(10,16)(12,21)(13,27)(14,22)(15,23)(17,29)(18,31)(19,30)(20,26)(24,39)(25,38)(28,37)(32,35)(33,36)(34,40);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,21)(13,28)(14,24)(15,25)(16,29)(18,33)(19,34)(20,35)(22,39)(23,38)(26,32)(27,37)(30,40)(31,36)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,d^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,c^b=cf,d^b=dh,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93979, #137305, #137306, … |
| maximal normal subgroup of | #93979, #137305, #137306, … |
| maximal quotient of | #121986, #122004, #122005, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4381);