Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #33830 |
| automorphism | Aut(G) order 8192 |
| 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 | #3334 | C2 |
| LCS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ G | |
| LECS | G ▷ C2×C8 ▷ C4 ▷ C2 ▷ C1 | |
251 conjugacy classes of subgroups; only the 101 normal subgroups are shown.
This group has 251 conjugacy classes of subgroups — too many to draw. Only its 101 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:
(1,2,5,4)(3,7,8,6)(9,10)(11,15)(12,16)(13,22)(14,17)(18,20)(19,23)(21,24);(1,3)(2,6)(4,7)(5,8)(9,11,14,13)(10,15,17,22)(12,19,21,20)(16,23,24,18);(1,4,5,2)(3,7,8,6)(10,16)(11,13)(12,20)(15,18)(17,24)(19,21)(22,23);(3,8)(6,7)(10,17)(15,22)(16,24)(18,23)Polycyclic presentation:
2^8|a^2=e,b^2=h,c^2=e,f^2=gh,g^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fg,g^a=gh,c^b=ch,d^b=de,f^c=fg,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #491937, #491942, #491944, … |
| maximal normal subgroup of | #491937, #491942, #491944, … |
| maximal quotient of | #497545, #497549, #497783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26908);