Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32875 |
| automorphism | Aut(G) order 24576 |
| 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 | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
297 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)(3,12)(4,16)(5,10)(6,21)(7,11)(8,15)(9,19)(13,28)(14,25)(17,26)(18,32)(20,27)(22,24)(23,30)(29,31);(1,3,6,22,7,15,5,14)(2,8,10,24,11,12,21,25)(4,17,20,13,19,29,18,30)(9,26,27,23,16,31,32,28);(1,4,6,20,7,19,5,18)(2,9,10,27,11,16,21,32)(3,13,14,17,15,30,22,29)(8,23,25,26,12,28,24,31);(1,5,7,6)(2,10,11,21)(3,14,15,22)(4,18,19,20)(8,24,12,25)(9,27,16,32)(13,29,30,17)(23,26,28,31)Polycyclic presentation:
2^8|b^2=eg,c^2=g,d^2=h,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 | #32877⋊C3 |
| maximal subgroup of | #144656, #144657, #144658, … |
| maximal normal subgroup of | #144656, #144657, #144658, … |
| maximal quotient of | #391516, #391529, #391530, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25865);