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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
748 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(11,15)(14,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,32)(26,30)(28,31);(1,2)(3,7)(4,5)(6,8)(9,10,13,12)(11,14,15,16)(18,22)(23,30)(24,27)(31,32);(2,5)(6,7)(9,11,13,15)(10,14,12,16)(17,19,21,26)(18,23,22,30)(20,28,29,25)(24,32,27,31);(1,2)(3,6)(4,5)(7,8)(17,20,21,29)(18,24,22,27)(19,25,26,28)(23,31,30,32);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,16,15,14)(17,18)(19,23)(20,24)(21,22)(25,31)(26,30)(27,29)(28,32)Polycyclic presentation:
2^8|b^2=g,c^2=fg,d^2=f,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×#41098, #7638961, #7639591, … |
| maximal normal subgroup of | C2×#41098, #7638961, #7639591, … |
| maximal quotient of | #544861, #545282, #545821, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34086);