Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #16249 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C24 | C24 |
| prime radical | C25 | C23 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
627 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)(6,9)(10,12)(13,14)(15,20)(16,19)(17,18)(22,25)(23,27);(6,9)(10,12)(14,18)(15,19)(21,22,24,25)(23,26,27,28);(1,2)(3,4)(5,6)(7,10)(8,9)(11,12)(13,15)(14,16)(17,19)(18,20)(21,22)(23,28)(24,25)(26,27);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,18)(17,20)(21,23)(22,26)(24,27)(25,28)Polycyclic presentation:
2^8|a^2=fg,b^2=h,d^2=eg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#17045, #612007, #612010, … |
| maximal normal subgroup of | C2×#17045, #612007, #612010, … |
| maximal quotient of | #155771, #155776, #155790, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 10033);