Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| 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 | |
356 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,22,19,21,24,30,26)(20,23,29,25,28,31,32,27);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,19,22,24,21,26,30,18)(20,27,29,23,28,25,32,31);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,21,28)(18,23,24,31)(19,25,26,27)(22,29,30,32);(2,5)(6,8)(11,16)(13,15)(18,24)(19,26)(20,28)(29,32)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=ef,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=ch,d^b=de,d^c=dh64 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23245, #6383127, #6383130, … |
| maximal normal subgroup of | C2×#23245, #6383127, #6383130, … |
| maximal quotient of | #356316, #356731, #356742, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16233);