Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #2127 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×Q8 | C23 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×Q8 ▷ C22 ▷ C2 ▷ C1 | |
256 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 256 conjugacy classes of subgroups — too many to draw. Only its 81 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)(6,11)(7,9)(8,16)(12,20)(13,19)(14,17);(5,6,9,13,10,14,7,12)(8,17,15,20,16,11,18,19);(1,3,4,2)(5,7,10,9)(6,12,14,13)(8,15,16,18)(11,19,17,20)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=eh,d^2=h,f^2=h,b^a=bd,c^a=ch,d^a=dh,f^a=fgh,d^b=dg,g^b=gh,f^d=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #130486, #130488, #130526, … |
| maximal normal subgroup of | #130486, #130488, #130526, … |
| maximal quotient of | #128313, #128339, #128371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5620);