Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1595 |
| automorphism | Aut(G) order 4096 |
| 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 | C2×C42 | C23 |
| socle | C22 | #379 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
486 conjugacy classes of subgroups; only the 87 normal subgroups are shown.
This group has 486 conjugacy classes of subgroups — too many to draw. Only its 87 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:
(1,2,4,6,5,8,3,7)(9,10,13,18,14,17,28,32)(11,19,22,33,21,15,26,30)(12,23,25,34,24,16,27,31)(20,35,37,40,36,29,38,39);(2,6)(3,4)(7,8)(9,11)(10,15)(12,20)(13,26)(14,21)(16,29)(17,19)(18,33)(22,28)(23,35)(24,36)(25,38)(27,37)(30,32)(31,40)(34,39);(9,12)(10,16)(11,20)(13,27)(14,24)(15,29)(17,23)(18,34)(19,35)(21,36)(22,38)(25,28)(26,37)(30,40)(31,32)(33,39)Polycyclic presentation:
2^8|a^2=f,d^2=h,f^2=gh,b^a=bd,c^a=ce,d^a=dg,e^a=eg,d^b=dh,e^b=eg,f^b=fgh,d^c=dg,f^c=fg46 conjugacy classes — view the full character table.
| maximal subgroup of | #137133, #137134, #137139, … |
| maximal normal subgroup of | #137133, #137134, #137139, … |
| maximal quotient of | #96950, #96951, #96958, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1397);