Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #8101 |
| automorphism | Aut(G) order 98304 |
| homology | Abelianization (H1) = C2×C42 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#95 |
| commutator | C23 | C2×C42 |
| Frattini, cosocle | C2×C42 | C23 |
| prime radical | C24 | C42 |
| LCS | G ▷ C23 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C2×C42 ▷ C22 ▷ C1 | |
260 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 3 generators on 32 points:
(1,2,5,8,6,9,4,13)(3,10,11,15,12,7,14,16)(17,18,20,25,22,24,21,29)(19,23,27,31,26,30,28,32);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,10)(13,16)(17,19,21,27,22,26,20,28)(18,23,25,32,24,30,29,31);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,10,16)(18,24)(23,30)(25,29)(31,32)Polycyclic presentation:
2^8|a^2=e,b^2=f,c^2=gh,e^2=g,f^2=h,b^a=bd,c^a=cg,d^a=dg,f^a=fh,d^b=dh,e^b=eg52 conjugacy classes — view the full character table.
| Fitting subgroup of | #8130⋊C3 |
| maximal subgroup of | #107212, #107399, #107400, … |
| maximal normal subgroup of | #107212, #107399, #107400, … |
| maximal quotient of | #95154, #95157, #95160, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 1118);