Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #14715 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
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 | |
689 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)(7,11)(10,12)(14,17)(15,19)(22,24)(23,27);(1,2)(3,4)(6,9)(10,12)(14,17)(18,20)(21,22,25,24)(23,26,27,28);(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,18,19,20)(21,23)(22,26)(24,28)(25,27);(5,7)(6,10)(8,11)(9,12)(13,15,16,19)(14,18,17,20)(21,24,25,22)(23,28,27,26)Polycyclic presentation:
2^8|b^2=gh,c^2=g,d^2=f,b^a=be,c^a=cf,d^a=dh,c^b=cg64 conjugacy classes — view the full character table.
| maximal subgroup of | #255110, #255130, #255140, … |
| maximal normal subgroup of | #255110, #255130, #255140, … |
| maximal quotient of | #154163, #154265, #154317, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 8088);