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 | C2×#496 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
480 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)(14,18)(15,20)(16,19);(1,2)(3,4)(6,9)(10,11)(13,14,16,18,17,15,19,20);(2,4)(13,15,16,20,17,14,19,18);(5,7,8,12)(6,10,9,11)(13,15,19,18,17,14,16,20)Polycyclic presentation:
2^8|b^2=f,c^2=f,d^2=eh,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×#26268, #6444510, #6444517, … |
| maximal normal subgroup of | C2×#26268, #6444510, #6444517, … |
| maximal quotient of | #307691, #307695, #307699, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19256);