Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23860 |
| 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×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
406 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)(7,10)(8,12)(9,11)(13,14,17,19)(15,22,23,28)(16,18,25,21)(20,26,27,24);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,18)(15,23)(16,25)(19,21)(20,28)(22,27);(2,4)(5,8,9,7)(6,10,11,12)(14,19)(18,21)(20,27)(22,28);(13,15,17,23)(14,20,19,27)(16,26,25,24)(18,22,21,28)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^2=fh,d^2=h,e^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,e^d=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26189, #6438424, #6438434, … |
| maximal normal subgroup of | C2×#26189, #6438424, #6438434, … |
| maximal quotient of | #306910, #306912, #306918, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19177);