Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8541 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C22×C4 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | #379 |
| commutator | C24 | C22×C4 |
| Frattini, prime radical, cosocle | C25 | C23 |
| LCS | G ▷ C24 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C25 ◁ G | |
| LECS | G ▷ C25 ▷ C22 ▷ C1 | |
455 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)(3,11,14,7)(4,16,17,8)(5,9)(6,10)(12,19)(13,27,28,20)(15,24,25,21)(18,32,26,22)(23,29,31,30);(1,3,12,25)(2,7,19,24)(4,13,26,28)(5,14,6,15)(8,20,32,27)(9,11,10,21)(16,31,22,23)(17,30,18,29);(1,4,5,18)(2,8,9,22)(3,13,14,29)(6,17,12,26)(7,20,11,23)(10,16,19,32)(15,30,25,28)(21,31,24,27)Polycyclic presentation:
2^8|a^2=f,b^2=gh,c^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,f^b=fg,f^c=fh40 conjugacy classes — view the full character table.
| Fitting subgroup of | #8545⋊C3 |
| maximal subgroup of | #93127, #93131, #93166, … |
| maximal normal subgroup of | #93127, #93131, #93166, … |
| maximal quotient of | #97068, #97076, #97086, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1533);