Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24390 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C22×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C22×#41 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
421 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)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,15,16)(17,18,21,26,22,27,20,25)(19,23,28,31,29,32,30,24);(1,3,4,8)(2,5,6,7)(9,11,12,15)(10,13,14,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,25)(26,31)(27,32);(11,15)(13,16)(17,19)(18,24)(20,28)(21,30)(22,29)(23,25)(26,32)(27,31);(2,6)(5,7)(17,20,22,21)(18,25,27,26)(19,28,29,30)(23,31,32,24)Polycyclic presentation:
2^8|a^2=fg,b^2=eg,d^2=h,f^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,d^b=dh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29119, #6461362, #6461370, … |
| maximal normal subgroup of | C2×#29119, #6461362, #6461370, … |
| maximal quotient of | #361744, #364359, #365141, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22107);