Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C2×#120 | C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
605 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6)(7,11)(8,10)(9,12)(13,14,17,19,16,15,18,20);(5,7)(6,9)(8,12)(10,11)(13,15,17,20,16,14,18,19);(2,4)(7,12)(9,11)(14,15)(19,20);(6,10)(9,11)Polycyclic presentation:
2^8|a^2=g,b^2=g,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=ceh76 conjugacy classes — view the full character table.
| maximal subgroup of | #122370, #122371, #128488, … |
| maximal normal subgroup of | #122370, #122371, #128488, … |
| maximal quotient of | #217612, #217613, #217614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 15018);