Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3514 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
469 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2,5,7)(3,9,10,16)(4,12,14,8)(6,13,15,11)(17,18,22,26)(19,27,30,31)(20,23,21,25)(24,29,32,28);(1,3)(2,6)(4,13)(5,10)(7,15)(8,16)(9,12)(11,14)(18,23)(20,21)(24,31)(25,26)(27,32)(28,29);(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,27)(26,32);(2,7)(3,10)(8,12)(11,13)(17,20,22,21)(18,25,26,23)(19,28,30,29)(24,27,32,31)Polycyclic presentation:
2^8|a^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18659, #6348249, #6348263, … |
| maximal normal subgroup of | C2×#18659, #6348249, #6348263, … |
| maximal quotient of | #166746, #166761, #166780, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11647);