Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #46350 |
| 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 | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
723 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(11,15)(14,16)(17,18,22,26)(19,27,28,23)(20,24,30,32)(21,25,31,29);(1,2)(3,5)(4,7)(6,8)(9,10,13,12)(11,14,15,16)(19,28)(21,31)(23,27)(25,29);(2,3)(6,7)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,25)(21,24)(22,28)(26,27)(29,30)(31,32);(1,3,5,2)(4,7,8,6)(17,20)(18,24)(19,29)(21,27)(22,30)(23,31)(25,28)(26,32);(1,4)(2,6)(3,7)(5,8)(9,12,13,10)(11,16,15,14)(17,21,22,31)(18,25,26,29)(19,24,28,32)(20,27,30,23)Polycyclic presentation:
2^8|a^2=h,b^2=gh,d^2=f,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#47359, #7701307, #7701443, … |
| maximal normal subgroup of | C2×#47359, #7701307, #7701443, … |
| maximal quotient of | #577018, #577019, #577147, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40347);