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 | #2671 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
776 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,12,14)(11,15,16,13)(17,18,21,22)(19,25,27,23)(20,28,29,24)(26,31,30,32);(1,2,5,6)(3,4,7,8)(11,16)(13,15)(18,22)(20,29)(23,25)(26,30);(2,6)(3,7)(9,11,12,16)(10,13,14,15)(17,19)(18,23)(20,30)(21,27)(22,25)(24,32)(26,29)(28,31);(1,3)(2,4)(5,7)(6,8)(9,10)(11,13)(12,14)(15,16)(17,20,21,29)(18,24,22,28)(19,26,27,30)(23,31,25,32);(1,4)(2,3)(5,8)(6,7)(17,18)(19,23)(20,24)(21,22)(25,27)(26,31)(28,29)(30,32)Polycyclic presentation:
2^8|a^2=g,b^2=fgh,c^2=fg,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×#54960, #8099728, #8100292, … |
| maximal normal subgroup of | C2×#54960, #8099728, #8100292, … |
| maximal quotient of | #585591, #585807, #586412, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 47948);