Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2812 |
| automorphism | Aut(G) = C213 (order 8192) · Inn(G) = C24 · Out(G) = C29 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C22×C4 | C24 |
| commutator, socle | C23 | C23×C4 |
| prime radical | C24 | C22×C4 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
303 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)(9,10,12,14)(11,13,16,15)(17,18,21,24,20,22,27,29)(19,23,26,30,25,28,31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(18,22)(23,28)(24,29)(30,32);(2,5)(6,7)(9,11,12,16)(10,13,14,15)(17,19,20,25)(18,23,22,28)(21,26,27,31)(24,30,29,32);(1,3,4,8)(2,6,5,7)(10,14)(13,15)(18,22)(19,25)(24,29)(26,31)Polycyclic presentation:
2^8|a^2=g,c^2=e,d^2=f,g^2=h,b^a=be,c^a=cf,d^a=def,c^b=cefh,d^b=deh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23382, #6407739, #6407741, … |
| maximal normal subgroup of | C2×#23382, #6407739, #6407741, … |
| maximal quotient of | #357315, #357339, #357343, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16370);