Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #99 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C4×C8 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C8 | C22⋊C4 |
| commutator | C22 | C2×C4×C8 |
| Frattini, cosocle | C22×C8 | C23 |
| socle | C22 | C2×C22⋊C8 |
| prime radical | C24 | C2×C8 |
| LCS | G ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C8 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
245 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 245 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 36 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 36 points:
(1,2,4,3)(5,6)(7,15)(8,19)(9,14)(10,12)(11,16)(13,21)(17,23)(18,28)(20,32)(22,25)(24,30)(26,33)(27,29)(31,34)(35,36);(5,7,8,17,9,18,20,31,10,16,21,29,22,30,33,36)(6,11,13,23,14,24,26,34,12,15,19,27,25,28,32,35);(1,3,4,2)(6,12)(11,15)(13,19)(14,25)(23,27)(24,28)(26,32)(34,35)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=e,f^2=g,g^2=h,b^a=bd,c^a=ch,f^a=fh,d^b=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #93286, #93287, #93414, … |
| maximal normal subgroup of | #93286, #93287, #93414, … |
| maximal quotient of | C8×#349, #128253, #128274, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5590);