Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24269 |
| automorphism | Aut(G) = C211×D4 (order 16384) · Inn(G) = C22×D4 · Out(G) = C29 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
375 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)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,31)(27,30)(28,32);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,25,28,21,27,20,26)(18,22,29,32,23,30,24,31);(10,14)(13,15)(17,19,21,27)(18,22,23,30)(20,28,25,26)(24,32,29,31);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(18,23)(22,30)(24,29)(31,32)Polycyclic presentation:
2^8|b^2=fgh,c^2=h,d^2=ef,g^2=h,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#29061, #6460393, #6460413, … |
| maximal normal subgroup of | C2×#29061, #6460393, #6460413, … |
| maximal quotient of | #362584, #363662, #367819, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22049);