Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6)(7,11)(8,9)(10,12)(13,14,19,16)(15,20,17,18);(1,2)(3,4)(6,9)(10,11)(13,15,18,14,19,17,20,16);(2,4)(13,16,18,15,19,14,20,17);(5,7,8,12)(6,10,9,11)(13,17,18,16,19,15,20,14)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=f,d^2=e,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fh,c^b=cg,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26257, #6444498, #6444511, … |
| maximal normal subgroup of | C2×#26257, #6444498, #6444511, … |
| maximal quotient of | #307418, #307422, #307426, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19245);