Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #17411 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×C22⋊C4 |
| commutator | C23 | C23×C4 |
| Frattini, cosocle | C24 | C24 |
| prime radical | C2×#548 | C2 |
| LCS | G ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×Q8 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
706 conjugacy classes of subgroups; only the 197 normal subgroups are shown.
This group has 706 conjugacy classes of subgroups — too many to draw. Only its 197 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,6)(7,8)(9,10)(11,15,16,13)(12,14);(1,2,5,6)(3,7,4,8);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15);(1,4,5,3)(2,7,6,8)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=e,d^2=e,b^a=be,c^a=cf,f^a=fh,d^b=de,g^c=gh55 conjugacy classes — view the full character table.
| Fitting subgroup of | #22534⋊C3 |
| maximal subgroup of | #139304, #139306, #139313, … |
| maximal normal subgroup of | #139304, #139306, #139313, … |
| maximal quotient of | #224489, #224491, #224493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15522);