Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2403 |
| 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 | |
366 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)(17,18,22,25,21,24,26,19)(20,28,31,27,30,23,32,29);(1,2)(3,6)(4,5)(7,8)(9,11,12,16)(10,13,14,15)(17,19,26,24,21,25,22,18)(20,29,32,23,30,27,31,28);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,20,21,30)(18,23,24,28)(19,27,25,29)(22,31,26,32);(2,5)(6,8)(11,16)(13,15)(20,30)(23,28)(27,29)(31,32)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=ef,g^2=h,b^a=be,c^a=cf,c^b=ch,d^b=de,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#23244, #6383127, #6383130, … |
| maximal normal subgroup of | C2×#23244, #6383127, #6383130, … |
| maximal quotient of | #356315, #356734, #356738, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16232);