Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53390 |
| 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 | |
745 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,12,14)(11,15,16,13)(17,18,22,27)(19,24,30,23)(20,25,26,21)(28,29,31,32);(1,2,5,3)(4,7,8,6)(11,16)(13,15)(17,19,22,30)(18,23,27,24)(20,31,26,28)(21,32,25,29);(2,3)(6,7)(9,11)(10,13)(12,16)(14,15)(17,19)(18,24)(20,31)(21,29)(22,30)(23,27)(25,32)(26,28);(1,3,5,2)(4,7,8,6)(9,10)(11,13)(12,14)(15,16)(17,20,22,26)(18,25,27,21)(19,28,30,31)(23,32,24,29);(1,4)(2,6)(3,7)(5,8)(17,21,22,25)(18,26,27,20)(19,29,30,32)(23,31,24,28)Polycyclic presentation:
2^8|a^2=fg,b^2=gh,d^2=gh,e^2=f,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#58272, #8104069, #8104120, … |
| maximal normal subgroup of | C2×#58272, #8104069, #8104120, … |
| maximal quotient of | #585534, #586292, #587180, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51260);