Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2010 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #127 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #399 |
| prime radical | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
307 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 307 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2)(3,4)(5,6,11,8,10,14,9,13)(7,15,18,12,19,20,17,16);(2,4)(5,7,11,17,10,19,9,18)(6,12,8,15,14,16,13,20);(2,4)(5,8,11,14,10,13,9,6)(7,16,17,20,19,12,18,15)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=de,d^2=h,f^2=h,g^2=h,b^a=bd,c^a=ce,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fh,g^b=gh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #93986, #137453, #137454, … |
| maximal normal subgroup of | #93986, #137453, #137454, … |
| maximal quotient of | #121624, #121670, #121671, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4363);