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 | |
767 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(9,10)(11,13)(12,15)(14,16)(17,18,22,23)(19,25,28,24)(20,29,31,21)(26,32,30,27);(1,2,5,4)(3,6,7,8)(10,11)(14,15)(18,23)(21,29)(24,25)(27,32);(1,3,5,7)(2,6,4,8)(9,11,13,10)(12,15,16,14)(17,19,22,28)(18,24,23,25)(20,30,31,26)(21,27,29,32);(9,12)(10,14)(11,15)(13,16)(17,20)(18,21)(19,26)(22,31)(23,29)(24,32)(25,27)(28,30);(1,4,5,2)(3,8,7,6)(17,21)(18,20)(19,27)(22,29)(23,31)(24,30)(25,26)(28,32)Polycyclic presentation:
2^8|a^2=fh,b^2=fg,c^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×#41742, #7639673, #7640012, … |
| maximal normal subgroup of | C2×#41742, #7639673, #7640012, … |
| maximal quotient of | #544825, #545265, #545783, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 34730);