Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32873 |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #2840 |
| commutator, Frattini, cosocle | C24 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
427 conjugacy classes of subgroups.
| minimal degree | μ(G) = 16 |
| transitive degree | μt(G) = 16 |
A minimal faithful permutation representation — 4 generators on 16 points:
(1,2)(3,9)(4,12)(5,8)(6,11)(7,13)(10,15)(14,16);(1,3)(2,6,8,9)(4,10)(5,11)(7,15,12,14)(13,16);(1,4)(2,7,8,12)(3,10)(5,13)(6,14,9,15)(11,16);(10,16)(14,15)Polycyclic presentation:
2^8|b^2=g,c^2=g,b^a=be,c^a=cf,g^a=gh,c^b=cg,d^b=def,e^b=eh,f^b=fh,d^c=de,f^c=fh28 conjugacy classes — view the full character table.
| Fitting subgroup of | #32873⋊C3, #72227533, #72227534 |
| maximal subgroup of | #144631, #144632, #144633, … |
| maximal normal subgroup of | #144631, #144632, #144633, … |
| maximal quotient of | #346276, #346277, #346278, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25861);