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 | #2635 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
780 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,4,5)(3,7,8,6)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31);(2,5)(3,8)(9,10)(11,12)(13,14)(15,16)(18,22)(19,25)(20,29)(30,32);(2,5)(6,7)(10,14)(11,15)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,18)(19,26)(20,24)(21,22)(23,25)(27,32)(28,29)(30,31);(1,3,4,8)(2,6,5,7)(9,12)(10,15)(11,14)(13,16)(17,20)(18,24)(19,27)(21,29)(22,28)(23,30)(25,31)(26,32)Polycyclic presentation:
2^8|a^2=g,c^2=fg,d^2=fgh,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×#58228, #8104596, #8104691, … |
| maximal normal subgroup of | C2×#58228, #8104596, #8104691, … |
| maximal quotient of | #585329, #586579, #586976, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 51216);