Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3003 |
| automorphism | Aut(G) = C23×#612424 (order 4096) · Inn(G) = C22×D4 · Out(G) = C24×D4 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | #3002 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C23×C8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
671 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,7,14,12)(8,11,16,19,17,15,20,18);(5,7)(6,10)(8,11)(9,12)(13,14)(15,17)(16,19)(18,20);(2,4);(5,8,9,16,10,17,14,20)(6,11,13,19,7,15,12,18)Polycyclic presentation:
2^8|a^2=g,d^2=g,e^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,d^b=dh,e^b=eh64 conjugacy classes — view the full character table.
| maximal subgroup of | #136816, #209032, #209034, … |
| maximal normal subgroup of | #136816, #209032, #209034, … |
| maximal quotient of | #208650, #208651, #208652, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14455);