Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32884 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
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 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
637 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,17,22)(15,23,25,19)(16,21,18,27)(20,26,28,24);(5,7,8,12)(6,9,10,11)(13,15)(14,19)(16,26)(17,25)(18,24)(20,21)(22,23)(27,28);(13,15)(14,20)(16,24)(17,25)(18,26)(19,21)(22,28)(23,27);(2,4)(13,16,17,18)(14,21,22,27)(15,24,25,26)(19,28,23,20)Polycyclic presentation:
2^8|a^2=e,b^2=eh,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=dg,f^a=fh,d^b=dh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6470469, #6470470, #6470472, … |
| maximal normal subgroup of | #6470469, #6470470, #6470472, … |
| maximal quotient of | #349339, #349341, #349343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25956);