Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| 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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
497 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,10)(7,11,12,9)(13,14)(15,22)(16,19)(17,21)(18,23)(20,26)(24,28)(25,27);(1,2)(3,4)(13,15,17,24)(14,18,21,27)(16,25,26,23)(19,22,20,28);(2,4)(5,7,8,12)(6,9,10,11);(14,19)(16,26)(18,22)(20,21)(23,25)(27,28)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=eh,e^d=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25954, #6438065, #6438086, … |
| maximal normal subgroup of | C2×#25954, #6438065, #6438086, … |
| maximal quotient of | #305248, #305250, #305276, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18942);