Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24289 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #2002 | C2 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
473 conjugacy classes of subgroups.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2,4,3)(5,6,9,12)(7,14,16,20)(8,13,17,11)(10,18,19,15);(1,3,4,2)(5,6,8,11,9,12,17,13)(7,10,15,14,16,19,18,20);(2,3)(5,6)(7,10)(8,11)(9,12)(13,17)(14,15)(16,19)(18,20);(5,7,8,15,9,16,17,18)(6,10,11,14,12,19,13,20)Polycyclic presentation:
2^8|a^2=f,b^2=efh,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#25431, #6444293, #6444299, … |
| maximal normal subgroup of | C2×#25431, #6444293, #6444299, … |
| maximal quotient of | #288028, #288034, #288039, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18419);