Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24274 |
| 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 | C22×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
435 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(5,6,8,9)(7,11,12,10)(13,14,17,19)(15,20,18,16);(1,2)(3,4)(6,9)(10,11)(13,14,16,18,17,19,20,15);(2,4)(13,14,16,18,17,19,20,15);(5,7,8,12)(6,10,9,11)(13,15,16,14,17,18,20,19)Polycyclic presentation:
2^8|a^2=ef,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^b=dh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26496, #6444669, #6444676, … |
| maximal normal subgroup of | C2×#26496, #6444669, #6444676, … |
| maximal quotient of | #309257, #309261, #309265, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19484);