Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1343 |
| automorphism | Aut(G) = #72283915 (order 4096) · Inn(G) = #99 · Out(G) = C22×AGL(1,ℤ/8) |
| homology | Abelianization (H1) = C42 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #99 |
| commutator | C22×C4 | C42 |
| Frattini, cosocle | C2×C4×Q8 | C22 |
| socle | C22 | #342 |
| prime radical | C23 | #95 |
| LCS | G ▷ C22×C4 ▷ C23 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C22×C4 ◁ C2×C4×Q8 ◁ G | |
| LECS | G ▷ C2×C4×Q8 ▷ C23 ▷ C2 ▷ C1 | |
125 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2,4,9,6,10,5,8)(3,11,12,7,14,15,13,16)(18,22)(19,25)(23,31)(24,30)(26,27)(29,32);(1,3,5,13,6,14,4,12)(2,7,9,15,10,16,8,11)(17,18,20,23,21,24,28,32)(19,22,26,29,25,30,27,31)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=f,d^2=fh,e^2=f,b^a=bc,c^a=cf,e^a=efg,c^b=cg,g^b=gh,e^c=eh46 conjugacy classes — view the full character table.
| maximal subgroup of | #93225, #93226, #93235, … |
| maximal normal subgroup of | #93225, #93226, #93235, … |
| maximal quotient of | #93399, #93426, #93573, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 274);