Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #576 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C22×C42 · Schur multiplier (H2) = C25×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | C22×D4 |
| commutator | C4 | C22×C42 |
| Frattini, cosocle | C22×C4 | C24 |
| socle | C22 | C2×C4×D4 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×M16 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,9,13,10,14,8,11)(7,12,17,20,15,19,16,18);(2,4)(6,11,14,13)(12,18,19,20);(5,7,9,17,10,15,8,16)(6,12,13,20,14,19,11,18);(7,15)(12,19)(16,17)(18,20)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=f,e^2=h,f^2=h,b^a=be,e^a=eh,g^a=gh,d^c=dh88 conjugacy classes — view the full character table.
| maximal subgroup of | #355822, #355824, #356068, … |
| maximal normal subgroup of | #355822, #355824, #356068, … |
| maximal quotient of | #182293, #182297, #182300, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13409);