Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C22×D4 | C2×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,4)(6,10)(11,12)(13,14,18,15,17,19,16,20);(5,6)(7,11)(8,12)(9,10)(13,15,18,19,17,20,16,14);(1,2,3,4)(5,7,9,8)(6,11,10,12)(14,19)(15,20);(5,8,9,7)(6,11,10,12)(13,16,17,18)(14,15,19,20)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=ef,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch,d^b=de64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23207, #6382832, #6382835, … |
| maximal normal subgroup of | C2×#23207, #6382832, #6382835, … |
| maximal quotient of | #231946, #231947, #231948, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16195);