Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of D4×D16 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C22×D8 |
| commutator, Frattini, cosocle | C2×C8 | C24 |
| prime radical | C2×D8 | C23 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×Q8 ◁ C8×Q8 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
253 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 253 conjugacy classes of subgroups — too many to draw. Only its 109 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(6,7)(9,10,13,17)(11,14,20,23)(12,21,19,16)(15,24,22,18);(1,2,4,5)(3,7,8,6);(10,14)(11,18)(12,19)(15,16)(17,23)(20,24)(21,22);(1,3,4,8)(2,6,5,7)Polycyclic presentation:
2^8|a^2=h,b^2=e,d^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,f^a=fg,g^a=gh,d^b=de,f^c=fg,g^c=gh55 conjugacy classes — view the full character table.
| Fitting subgroup of | #33926⋊C3 |
| maximal subgroup of | #491777, #491784, #491793, … |
| maximal normal subgroup of | #491777, #491784, #491793, … |
| maximal quotient of | Q8×#357, #497666, #497954, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26914);