Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of H(3,ℤ/4) |
| automorphism | Aut(G) order 24576 |
| homology | Abelianization (H1) = C82 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C42 |
| commutator | C4 | C82 |
| Frattini, cosocle | C43 | C22 |
| socle, prime radical | C23 | #95 |
| LCS | G ▷ C4 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C43 ▷ C23 ▷ C1 | |
169 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,3,5,4,6,7,8)(17,18)(19,25,27,31)(20,23)(21,24)(22,26,29,32)(28,30);(9,10,11,13,12,14,15,16)(17,19)(18,22,24,29)(20,26)(21,27)(23,25,30,31)(28,32)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=g,e^2=h,b^a=bc,e^a=ef,d^b=df88 conjugacy classes — view the full character table.
| derived subgroup of | #7014⋊C3 |
| Fitting subgroup of | #7014⋊C3 |
| maximal subgroup of | #92837, #92838, C2×#7014, … |
| maximal normal subgroup of | #92837, #92838, C2×#7014, … |
| maximal quotient of | #92806, #93249, #93250, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 2);