Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #10690 |
| automorphism | Aut(G) order 12288 |
| homology | Abelianization (H1) = C23 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #401 |
| commutator, Frattini, prime radical, cosocle | C25 | C23 |
| LCS | G ▷ C25 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
186 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2)(3,12,15,23)(4,16,31,25)(5,9)(6,11)(7,14,22,30)(8,18,26,19)(10,20)(13,27,28,24)(17,21,29,32);(1,3)(2,7,11,23)(4,17,18,28)(5,15)(6,14)(8,24,26,32)(9,22,10,12)(13,19,29,31)(16,27,25,21)(20,30);(1,4)(2,8,9,25)(3,13,14,28)(5,18)(6,19)(7,21,12,27)(10,26,11,16)(15,29,30,17)(20,31)(22,24,23,32)Polycyclic presentation:
2^8|a^2=def,b^2=de,c^2=d,b^a=bd,c^a=ce,d^a=dg,e^a=egh,f^a=fh,c^b=cf,d^b=dh,e^b=eh,f^b=fg,e^c=eg,f^c=fh25 conjugacy classes — view the full character table.
| Fitting subgroup of | #11709159 |
| maximal subgroup of | C2×#10690, #244495, #10690⋊C2, … |
| maximal normal subgroup of | C2×#10690, #244495, #10690⋊C2, … |
| maximal quotient of | #121068, #121069, #146290, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 3678);