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 | C2×#120 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
795 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,12,14)(11,15,16,13)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31);(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,13,16,15)(18,22)(19,25)(20,29)(30,32);(1,2,4,6)(3,5,8,7)(9,10)(11,13)(12,14)(15,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,2)(3,5)(4,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32);(1,3)(2,5)(4,8)(6,7)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=fgh,d^2=fg,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×#56491, #8099012, #8099098, … |
| maximal normal subgroup of | C2×#56491, #8099012, #8099098, … |
| maximal quotient of | #586109, #587152, #587159, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 49479);