Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #1284 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C4×C8 · Schur multiplier (H2) = C2×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C2×C4 | #95 |
| commutator | C2×C4 | C4×C8 |
| Frattini, cosocle | C2×C4×C8 | C22 |
| socle | C22 | #336 |
| prime radical | C23 | C22⋊C8 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ C2×C4×C8 ◁ G | |
| LECS | G ▷ C2×C4×C8 ▷ C2×C4 ▷ C2 ▷ C1 | |
117 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,9,11,15)(4,7)(5,8)(6,10,13,16)(12,14)(17,18,19,23,21,24,26,30,20,22,25,28,27,29,31,32);(1,3)(2,6,8,13)(4,10)(5,11)(7,9,14,15)(12,16)(18,22)(19,25)(24,29)(26,31)Polycyclic presentation:
2^8|a^2=d,b^2=e,c^2=fh,d^2=g,g^2=h,b^a=bc,c^a=ch,e^a=efh,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | #92827, #92828, #92837, … |
| maximal normal subgroup of | #92827, #92828, #92837, … |
| maximal quotient of | #93250, #93288, #93304, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 131);