Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30396 |
| 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 | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
333 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)(3,5)(4,6)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,22)(20,23)(21,24)(25,32)(26,30)(27,29)(28,31);(1,3,5,2)(4,7,8,6)(9,11,12,15)(10,13,14,16)(17,19,20,26)(18,22,23,30)(21,25,28,27)(24,32,31,29);(1,4,5,8)(2,6,3,7)(10,14)(13,16)(18,23)(19,25)(21,28)(22,29)(26,27)(30,32);(2,3)(6,7)(10,14)(13,16)(18,23)(22,30)(24,31)(29,32)Polycyclic presentation:
2^8|a^2=fh,b^2=ef,c^2=e,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,g^b=gh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6634774, #6634787, #6634805, … |
| maximal normal subgroup of | #6634774, #6634787, #6634805, … |
| maximal quotient of | #383799, #383962, #384186, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24734);