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 | |
736 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,7)(3,8,4,6)(10,12)(11,13)(17,18,22,25)(19,20,23,28)(21,24,30,32)(26,27,29,31);(1,3)(2,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,23)(18,20,25,28)(21,29,30,26)(24,31,32,27);(3,4)(6,8)(9,11,14,13)(10,15,12,16)(17,19)(18,20)(21,26)(22,23)(24,27)(25,28)(29,30)(31,32);(1,4,5,3)(2,6,7,8)(9,12,14,10)(11,15,13,16)(17,20)(18,23)(19,25)(21,27)(22,28)(24,29)(26,32)(30,31);(1,2)(3,8)(4,6)(5,7)(9,13,14,11)(10,15,12,16)(17,21)(18,24)(19,26)(20,27)(22,30)(23,29)(25,32)(28,31)Polycyclic presentation:
2^8|a^2=h,b^2=fgh,c^2=g,d^2=f,e^2=g,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×#56071, #8103042, #8103199, … |
| maximal normal subgroup of | C2×#56071, #8103042, #8103199, … |
| maximal quotient of | #585585, #586208, #587013, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49059);