Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C27 |
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 | C22×C22⋊C4 |
| prime radical | #3449 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×#41 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
891 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)(6,10,12,15)(11,17,18,20);(5,6,8,15,9,12,16,10)(7,11,14,20,13,18,19,17);(2,4)(5,7)(6,11)(8,14)(9,13)(10,17)(12,18)(15,20)(16,19);(7,13)(11,18)(14,19)(17,20)Polycyclic presentation:
2^8|a^2=g,b^2=eg,e^2=h,b^a=be,c^a=cf,e^b=eh,g^b=gh,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | #147522, #147527, #148017, … |
| maximal normal subgroup of | #147522, #147527, #148017, … |
| maximal quotient of | #193427, #193428, #193429, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13891);