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×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
490 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)(13,14,18,22)(15,23,26,20)(16,21,17,19)(24,27,25,28);(1,2)(3,4)(5,7,8,11)(6,9,10,12)(14,19)(16,17)(20,27)(21,22)(23,28)(24,25);(2,4)(6,10)(9,12)(13,15,18,26)(14,20,22,23)(16,24,17,25)(19,27,21,28);(13,16)(15,24)(17,18)(19,21)(25,26)(27,28)Polycyclic presentation:
2^8|a^2=h,b^2=fh,c^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×#24923, #6438199, #6438207, … |
| maximal normal subgroup of | C2×#24923, #6438199, #6438207, … |
| maximal quotient of | #280415, #280417, #280423, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17911);