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 | |
493 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)(7,9)(8,10)(11,12)(13,14,17,21)(15,19,23,27)(16,18,24,20)(22,25,28,26);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,18)(16,24)(19,25)(20,21)(22,28)(26,27);(2,4)(6,10)(9,12)(13,15,17,23)(14,19,21,27)(16,22,24,28)(18,25,20,26);(14,18)(16,24)(19,25)(20,21)(22,28)(26,27)Polycyclic presentation:
2^8|a^2=h,b^2=f,c^2=h,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×#24903, #6438182, #6438201, … |
| maximal normal subgroup of | C2×#24903, #6438182, #6438201, … |
| maximal quotient of | #279887, #279889, #279895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17891);