Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2846 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C2×#120 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
854 conjugacy classes of subgroups; only the 223 normal subgroups are shown.
This group has 854 conjugacy classes of subgroups — too many to draw. Only its 223 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:
(1,2)(3,4)(5,6)(7,8)(9,10,12,11,13,14,16,15);(2,4);(6,8);(9,11,12,14,13,15,16,10)Polycyclic presentation:
2^8|a^2=g,d^2=g,g^2=h,b^a=be,c^a=cf,d^a=dh88 conjugacy classes — view the full character table.
| Fitting subgroup of | #21440⋊C3, #72342671, #72342672 |
| maximal subgroup of | #208806, #208822, #265267, … |
| maximal normal subgroup of | #208806, #208822, #265267, … |
| maximal quotient of | #208668, #208669, #208670, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14428);