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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
784 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)(3,5)(4,7)(6,8)(9,10,13,16)(11,14,12,15)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27);(2,3)(6,7)(9,11,13,12)(10,14,16,15)(18,23)(21,29)(24,25)(27,32);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,15)(13,16)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,11)(10,15,16,14)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30);(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,c^2=f,d^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×#41515, #7639673, #7639866, … |
| maximal normal subgroup of | C2×#41515, #7639673, #7639866, … |
| maximal quotient of | #544653, #545177, #545625, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34503);