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 | C22×#41 | C22 |
| LCS | G ▷ C24 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C24 ◁ G | |
| LECS | G ▷ C24 ▷ C2 ▷ C1 | |
344 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,11)(4,15)(5,9)(6,10)(7,14)(8,17)(12,21)(13,24)(16,29)(18,23)(19,26)(20,28)(22,30)(25,32)(27,31);(1,3)(2,7,10,11)(4,12)(5,13)(6,14)(8,21,15,19)(9,20,23,24)(16,27)(17,26)(18,28)(22,31,29,32)(25,30);(1,4)(2,8,10,15)(3,12)(5,16)(6,17)(7,19,11,21)(9,22,23,29)(13,27)(14,26)(18,30)(20,32,24,31)(25,28);(1,5,6,18)(2,9,10,23)(3,13,14,28)(4,16,17,30)(7,20,11,24)(8,22,15,29)(12,25,26,27)(19,31,21,32)Polycyclic presentation:
2^8|b^2=g,c^2=g,d^2=h,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 | #32874⋊C3 |
| maximal subgroup of | #144635, #144636, #144637, … |
| maximal normal subgroup of | #144635, #144636, #144637, … |
| maximal quotient of | #346277, #346278, #346279, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25862);