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) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C24 | C24 |
| LCS | G ▷ C24 ▷ C1 | |
| UCS | C1 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C1 | |
543 conjugacy classes of subgroups.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 28 points:
(1,2)(3,4)(7,11)(10,12)(13,14,16,18)(15,19,20,17)(21,22,25,28)(23,26,24,27);(1,2,4,3)(6,9)(10,12)(21,23,25,24)(22,26,28,27);(2,3)(5,6,8,9)(7,10,11,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,26)(24,27)(25,28);(1,3,4,2)(5,7)(6,10)(8,11)(9,12)(13,15,16,20)(14,17,18,19)(21,24,25,23)(22,27,28,26)Polycyclic presentation:
2^8|a^2=ef,b^2=fgh,c^2=eg,d^2=f,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×#16999, #611884, #611894, … |
| maximal normal subgroup of | C2×#16999, #611884, #611894, … |
| maximal quotient of | #155761, #155763, #155817, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 9987);