Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #1355 |
| automorphism | Aut(G) = #72279639 (order 2048) · Inn(G) = #329 · Out(G) = #120 |
| homology | Abelianization (H1) = C2×C8 · Schur multiplier (H2) = C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C4 | #329 |
| commutator | C42 | C2×C8 |
| Frattini, cosocle | #376 | C22 |
| socle | C2 | #1272 |
| prime radical | C23 | #99 |
| LCS | G ▷ C42 ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C4 ◁ C22×C4 ◁ #376 ◁ G | |
| LECS | G ▷ #376 ▷ C22×C4 ▷ C2 ▷ C1 | |
105 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 2 generators on 32 points:
(1,2)(3,11,13,26,14,27,19,30)(4,15,17,31)(5,9)(6,10)(7,12,21,28,23,16,22,29)(8,18,24,32)(20,25);(1,3,6,14)(2,7)(4,16,17,12)(5,19,20,13)(8,11)(9,21)(10,23)(15,30)(18,29,32,28)(22,25)(24,27)(26,31)Polycyclic presentation:
2^8|a^2=d,b^2=f,c^2=fh,d^2=g,e^2=h,b^a=bc,c^a=ce,f^a=fh,c^b=cf,d^b=def,e^b=eh,g^b=gh,d^c=dh40 conjugacy classes — view the full character table.
| maximal subgroup of | #94546, #94602, C2×#7129, … |
| maximal normal subgroup of | #94546, #94602, C2×#7129, … |
| maximal quotient of | #92885, #92889, #92890, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 117);