Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32875 |
| automorphism | Aut(G) order 49152 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2842 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(2,6)(3,9)(4,12)(7,14);(1,2,4,13,5,6,12,8)(3,10,11,7,9,15,16,14);(1,3,4,11,5,9,12,16)(2,7,8,10,6,14,13,15);(7,14)(10,15)Polycyclic presentation:
2^8|b^2=eg,c^2=g,g^2=h,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=def,e^b=eh,f^b=fh,g^b=gh,d^c=de,f^c=fh28 conjugacy classes — view the full character table.
| Fitting subgroup of | #32875⋊C3 |
| maximal subgroup of | #144648, #144649, #144652, … |
| maximal normal subgroup of | #144648, #144649, #144652, … |
| maximal quotient of | #391515, #391529, #392034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25863);