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 | #392 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
765 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,4)(3,7,8,6)(10,12)(11,15)(17,18)(19,24)(20,21)(22,23)(25,30)(26,32)(27,29)(28,31);(1,2,5,4)(3,6,8,7)(9,10,13,12)(11,16,15,14)(18,23)(21,25)(24,31)(27,32);(1,3,5,8)(2,6,4,7)(9,11,13,15)(10,14,12,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,31)(25,32)(26,30);(1,2,5,4)(3,6,8,7)(9,12,13,10)(11,16,15,14)(17,20)(18,25)(19,26)(21,23)(22,30)(24,27)(28,29)(31,32);(1,4,5,2)(3,7,8,6)(17,21,22,25)(18,20,23,30)(19,27,28,32)(24,29,31,26)Polycyclic presentation:
2^8|a^2=gh,b^2=g,c^2=g,d^2=g,e^2=fg,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×#41923, #7639909, #7640250, … |
| maximal normal subgroup of | C2×#41923, #7639909, #7640250, … |
| maximal quotient of | #544905, #545309, #545953, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34911);