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 | |
755 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,8)(3,6,4,7)(10,12)(14,15)(17,18)(19,26)(20,21)(22,23)(24,29)(25,31)(27,28)(30,32);(1,3,5,4)(2,6,8,7)(9,10,13,12)(11,15,16,14)(18,23)(21,25)(24,26)(28,32);(1,2)(3,6)(4,7)(5,8)(9,11,13,16)(10,14,12,15)(17,19,22,29)(18,24,23,26)(20,30,31,27)(21,28,25,32);(1,3,5,4)(2,6,8,7)(9,12,13,10)(11,15,16,14)(17,20)(18,25)(19,27)(21,23)(22,31)(24,28)(26,32)(29,30);(1,4,5,3)(2,7,8,6)(17,21,22,25)(18,20,23,31)(19,28,29,32)(24,30,26,27)Polycyclic presentation:
2^8|a^2=fgh,b^2=fg,c^2=f,d^2=fg,e^2=g,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×#41847, #7639886, #7640248, … |
| maximal normal subgroup of | C2×#41847, #7639886, #7640248, … |
| maximal quotient of | #544919, #545315, #545972, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34835);