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×D4 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
403 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,3)(5,6,8,9)(7,11,12,10)(13,14)(15,21)(16,19)(17,20)(18,23)(22,26)(24,28)(25,27);(1,2,4,3)(6,9)(10,11)(13,15)(14,18)(16,23)(17,22)(19,21)(20,28)(24,25)(26,27);(1,2,4,3)(14,19)(15,22)(17,25)(18,26)(21,28)(23,24);(1,3,4,2)(5,7,8,12)(6,10,9,11)(14,19)(18,21)(20,27)(26,28)Polycyclic presentation:
2^8|a^2=ef,b^2=fh,c^2=fh,d^2=eh,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26471, #6444594, #6444604, … |
| maximal normal subgroup of | C2×#26471, #6444594, #6444604, … |
| maximal quotient of | #308977, #308978, #308979, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19459);