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×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#99 |
| commutator | C2×C4 | C2×C42 |
| Frattini, cosocle | C22×D4 | C23 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C2×C4 ▷ C22 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ G | |
| LECS | G ▷ C22×D4 ▷ C22 ▷ C2 ▷ C1 | |
314 conjugacy classes of subgroups; only the 81 normal subgroups are shown.
This group has 314 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) = 32 |
A minimal faithful permutation representation — 3 generators on 20 points:
(1,2,4,3)(5,6,9,11)(7,13,15,20)(8,16,17,12)(10,18,19,14);(5,7)(6,10)(8,14,17,18)(9,15)(11,19)(12,13,16,20);(1,3,4,2)(6,11)(10,19)(12,16)(13,20)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=eh,d^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 | #130482, #130484, #130523, … |
| maximal normal subgroup of | #130482, #130484, #130523, … |
| maximal quotient of | #128313, #128338, #128371, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 5616);