Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18712 |
| 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 | #3212 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
426 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,9)(4,6)(5,13)(7,11)(8,15)(10,14)(12,16)(18,23)(19,26)(20,21)(25,29)(27,28)(30,31);(1,3,6,12)(2,7,4,10)(5,14,15,11)(8,16,13,9)(17,18,22,25)(19,24,28,32)(20,23,21,29)(26,30,27,31);(1,4,6,2)(3,10,12,7)(5,13,15,8)(9,11,16,14)(17,19,22,28)(18,24,25,32)(20,26,21,27)(23,30,29,31);(2,4)(7,10)(8,13)(9,16)(17,20,22,21)(18,23,25,29)(19,27,28,26)(24,31,32,30)Polycyclic presentation:
2^8|b^2=g,c^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,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#18744, #6348357, #6348359, … |
| maximal normal subgroup of | C2×#18744, #6348357, #6348359, … |
| maximal quotient of | #168145, #168180, #168227, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11732);