Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #17082 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
733 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(2,4)(5,6)(7,11)(8,9)(10,12)(13,14,16,17)(15,18,20,19)(21,22)(23,25)(24,27)(26,28);(6,9)(7,12)(14,17)(18,19)(21,23,24,28)(22,25,27,26);(5,6)(7,11)(8,9)(10,12)(13,15)(14,18)(16,20)(17,19)(21,23)(22,26)(24,28)(25,27);(1,2)(3,4)(5,7)(6,10)(8,12)(9,11)(13,14)(15,19)(16,17)(18,20)(21,22)(23,26)(24,27)(25,28)Polycyclic presentation:
2^8|a^2=efg,b^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=def,d^c=de46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17146, #844933, #844943, … |
| maximal normal subgroup of | C2×#17146, #844933, #844943, … |
| maximal quotient of | #156472, #156481, #156504, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10134);