Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #447 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C2×D4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| socle | C23 | C2×C22⋊C4 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
357 conjugacy classes of subgroups; only the 175 normal subgroups are shown.
This group has 357 conjugacy classes of subgroups — too many to draw. Only its 175 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2)(3,4)(6,10)(7,8)(11,12)(13,14,17,20,18,21,16,26)(15,22,25,27,23,28,24,19);(2,4)(5,6,8,11,9,12,7,10)(13,15)(14,19,21,27)(16,24)(17,25)(18,23)(20,22,26,28);(15,23)(19,27)(22,28)(24,25)Polycyclic presentation:
2^8|a^2=e,b^2=f,d^2=g,e^2=h,f^2=gh,b^a=bd,d^a=dg,f^a=fg,c^b=ch76 conjugacy classes — view the full character table.
| maximal subgroup of | #106178, #106179, #106732, … |
| maximal normal subgroup of | #106178, #106179, #106732, … |
| maximal quotient of | #94885, #94901, #94906, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 881);