Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
405 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 405 conjugacy classes of subgroups — too many to draw. Only its 171 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) = 256 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(6,10)(11,12)(14,17)(15,19)(18,20);(1,3,4,2)(5,6)(7,11)(8,12)(9,10)(14,18)(17,20);(5,7,9,8)(6,11,10,12)(13,14,19,20,16,18,15,17);(5,8,9,7)(6,11,10,12)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=efh,d^2=e,f^2=h,b^a=be,c^a=cf,f^a=fh,c^b=ch,d^b=de58 conjugacy classes — view the full character table.
| maximal subgroup of | #229322, #229324, #229330, … |
| maximal normal subgroup of | #229322, #229324, #229330, … |
| maximal quotient of | #231858, #231870, #231882, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16210);