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:
(1,2,6,7)(3,5,4,8)(11,15)(13,16)(17,18)(19,25)(20,22)(21,28)(23,27)(24,29)(26,30)(31,32);(2,7)(3,4)(9,10,12,14)(11,13,15,16)(18,20)(21,29)(23,25)(26,32);(1,3,6,4)(2,8,7,5)(9,11,12,15)(10,13,14,16)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,4,6,3)(2,5,7,8)(9,11)(10,13)(12,15)(14,16)(17,20,22,18)(19,23,27,25)(21,28,29,24)(26,30,32,31);(1,5)(2,3)(4,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=g,d^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×#54059, #8099053, #8099158, … |
| maximal normal subgroup of | C2×#54059, #8099053, #8099158, … |
| maximal quotient of | #585703, #585707, #586367, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47047);