Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| 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×D8 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
414 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)(7,11)(8,9)(10,12)(13,14,16,21,17,20,25,19)(15,18,23,27,24,26,22,28);(1,2)(3,4)(6,9)(10,11)(13,15,16,23,17,24,25,22)(14,18,21,27,20,26,19,28);(2,4)(14,19)(15,22)(16,25)(18,26)(20,21)(23,24);(5,7,8,12)(6,10,9,11)(14,20)(18,26)(19,21)(27,28)Polycyclic presentation:
2^8|a^2=f,b^2=f,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25298, #6448098, #6448108, … |
| maximal normal subgroup of | C2×#25298, #6448098, #6448108, … |
| maximal quotient of | #286795, #286801, #286807, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18286);