Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of D42 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator | C22 | C24×C4 |
| Frattini, socle, cosocle | C23 | C25 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1199 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 20 points:
(1,2,4,3)(6,9)(10,12)(13,14)(15,20)(16,18)(17,19);(1,3,4,2)(5,6)(7,10)(8,9)(11,12);(13,15,17,16)(14,18,19,20);(6,9)(10,12)(13,16,17,15)(14,18,19,20);(5,7,8,11)(6,10,9,12)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,d^b=df100 conjugacy classes — view the full character table.
| maximal subgroup of | #106006, #182286, #182289, … |
| maximal normal subgroup of | #106006, #182286, #182289, … |
| maximal quotient of | #513466, #513489, #513493, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 27139);