Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #2018 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #121 |
| commutator | C22×C4 | C22×C4 |
| Frattini, cosocle | C23×C4 | C23 |
| socle | C22 | #388 |
| prime radical | C24 | #41 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C23×C4 ▷ C2 ▷ C1 | |
289 conjugacy classes of subgroups; only the 103 normal subgroups are shown.
This group has 289 conjugacy classes of subgroups — too many to draw. Only its 103 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6,9,15,10,14,20,13)(7,8,17,18,12,19,16,11);(1,3,4,2)(5,7,10,12)(6,11,14,18)(8,13,19,15)(9,17,20,16);(2,3)(5,8,9,18,10,19,20,11)(6,12,13,17,14,7,15,16)Polycyclic presentation:
2^8|a^2=g,b^2=de,c^2=d,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#11304, #358340, #358412, … |
| maximal normal subgroup of | C2×#11304, #358340, #358412, … |
| maximal quotient of | #121078, #121092, #121093, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 4292);