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 | C24 | C2×Q8 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
292 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,6,10)(3,11,13,7)(4,15,17,8)(5,19,21,9)(12,27,28,22)(14,23,20,26)(16,24,31,29)(18,25,32,30);(1,3,5,20,6,13,21,14)(2,7,9,26,10,11,19,23)(4,16,18,12,17,31,32,28)(8,24,25,22,15,29,30,27);(1,4,5,18,6,17,21,32)(2,8,9,25,10,15,19,30)(3,12,14,16,13,28,20,31)(7,22,23,24,11,27,26,29);(12,28)(16,31)(22,27)(24,29)Polycyclic presentation:
2^8|a^2=h,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 | #32876⋊C3 |
| maximal subgroup of | #144650, #144651, #144653, … |
| maximal normal subgroup of | #144650, #144651, #144653, … |
| maximal quotient of | #391515, #391530, #392035, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 25864);