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 | |
738 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)(17,18,22,23)(19,24,28,30)(20,27,31,25)(21,26,32,29);(1,2)(3,7)(4,5)(6,8)(9,10,13,12)(11,14,15,16)(18,23)(21,32)(24,30)(25,27);(2,5)(6,7)(9,11,13,15)(10,14,12,16)(17,19)(18,24)(20,29)(21,27)(22,28)(23,30)(25,32)(26,31);(1,2)(3,6)(4,5)(7,8)(17,20,22,31)(18,25,23,27)(19,26,28,29)(21,30,32,24);(1,3)(2,6)(4,8)(5,7)(9,12,13,10)(11,16,15,14)(17,21)(18,26)(19,27)(20,30)(22,32)(23,29)(24,31)(25,28)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,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×#41095, #7638930, #7639591, … |
| maximal normal subgroup of | C2×#41095, #7638930, #7639591, … |
| maximal quotient of | #544843, #545270, #545803, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34083);