Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
369 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,8,9)(7,11,12,10)(13,14,18,23,19,24,17,22)(15,20,25,21,16,26,27,28);(1,2)(3,4)(6,9)(10,11)(13,15)(14,20)(16,19)(17,27)(18,25)(21,23)(22,28)(24,26);(2,4)(13,16,19,15)(14,21,24,28)(17,27,18,25)(20,22,26,23);(5,7,8,12)(6,10,9,11)(13,17,19,18)(14,22,24,23)(15,25,16,27)(20,21,26,28)Polycyclic presentation:
2^8|a^2=ef,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#24715, #6449900, #6449906, … |
| maximal normal subgroup of | C2×#24715, #6449900, #6449906, … |
| maximal quotient of | #276704, #276706, #276712, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17703);