Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32878 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2856 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
508 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 20 points:
(2,3)(5,6)(7,9)(8,10)(11,12)(14,17)(15,18)(19,20);(1,2,4,3)(5,7,8,11)(6,9,10,12);(6,10)(9,12)(13,14,16,17)(15,18,19,20);(1,3,4,2)(13,15)(14,18)(16,19)(17,20)Polycyclic presentation:
2^8|b^2=efh,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=dg,g^a=gh,g^d=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #100240, #100242, #100256, … |
| maximal normal subgroup of | #100240, #100242, #100256, … |
| maximal quotient of | #351294, #351295, #351296, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26101);