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 | #2673 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
838 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)(4,6,8,7)(17,18)(19,20)(21,29)(22,23)(24,27)(25,30)(26,31)(28,32);(1,2,5,3)(4,7,8,6)(9,10,13,15)(11,12,16,14)(18,23)(20,24)(25,29)(28,31);(2,3)(6,7)(9,11,13,16)(10,14,15,12)(17,19)(18,24)(20,23)(21,26)(22,27)(25,31)(28,29)(30,32);(1,3,5,2)(4,7,8,6)(9,11)(10,12)(13,16)(14,15)(17,20,22,24)(18,19,23,27)(21,28,30,31)(25,26,29,32);(1,4)(2,6)(3,7)(5,8)(9,12)(10,11)(13,14)(15,16)(17,21,22,30)(18,25,23,29)(19,26,27,32)(20,28,24,31)Polycyclic presentation:
2^8|a^2=fg,b^2=h,c^2=fgh,d^2=g,e^2=f,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×#47627, #7701577, #7701592, … |
| maximal normal subgroup of | C2×#47627, #7701577, #7701592, … |
| maximal quotient of | #576814, #576883, #577037, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 40615);