Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2204 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | B(4,𝔽2) |
| commutator | C2×Q8 | C22×C4 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C23 | #120 |
| LCS | G ▷ C2×Q8 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
230 conjugacy classes of subgroups; only the 59 normal subgroups are shown.
This group has 230 conjugacy classes of subgroups — too many to draw. Only its 59 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:
(1,2,4,3)(5,6)(7,13)(8,11)(9,18)(10,15)(12,16)(14,19)(17,20);(1,3,4,2)(6,11)(7,8)(9,19)(12,14)(13,15)(18,20);(5,7,10,8)(6,12,15,20)(9,17,19,16)(11,18,13,14)Polycyclic presentation:
2^8|a^2=f,b^2=f,c^2=h,d^2=h,e^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gh,c^b=ch,d^b=dh,e^b=egh,d^c=dg,e^c=eh,e^d=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #152465, #152466, #152471, … |
| maximal normal subgroup of | #152465, #152466, #152471, … |
| maximal quotient of | #135945, #135965, #135979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5961);