Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1504 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C2×C22⋊C4 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C22⋊C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
331 conjugacy classes of subgroups; only the 127 normal subgroups are shown.
This group has 331 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) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 3 generators on 28 points:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,22)(16,25)(17,20)(18,21)(19,24)(23,28)(26,27);(6,10,12,11)(13,15,17,24,18,23,16,26)(14,19,20,28,21,27,25,22);(5,7,9,8)(6,11,12,10)(15,23)(19,27)(22,28)(24,26)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=gh,d^2=g,f^2=h,b^a=bd,d^a=dg,f^a=fgh,c^b=ch,d^b=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #105800, #105801, #106579, … |
| maximal normal subgroup of | #105800, #105801, #106579, … |
| maximal quotient of | #95268, #95274, #95278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1024);