Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) = C215 (order 32768) · Inn(G) = C25 · Out(G) = C210 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2676 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(5,6)(7,8)(17,18,21,20)(19,23,24,22);(2,6)(4,7)(9,10)(11,15)(12,13)(14,16)(17,18,21,20)(19,22,24,23);(2,6)(3,8)(10,13)(14,15)(17,19,21,24)(18,22,20,23);(1,3)(2,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,20,21,18)(19,23,24,22);(1,4,5,7)(2,3,6,8)(9,11)(10,14)(12,16)(13,15)Polycyclic presentation:
2^8|a^2=fg,b^2=fg,c^2=fg,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#40911, #7639568, #7639584, … |
| maximal normal subgroup of | C2×#40911, #7639568, #7639584, … |
| maximal quotient of | #544668, #545188, #545700, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33899);