Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #40484 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, prime radical, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
702 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,6,8,7)(11,15)(14,16)(17,18)(19,27)(20,29)(21,26)(22,23)(24,30)(25,31)(28,32);(1,2,5,3)(4,7,8,6)(9,10,13,12)(11,14,15,16)(18,23)(21,32)(24,27)(25,29);(2,3)(6,7)(9,11)(10,14)(12,16)(13,15)(17,19,22,30)(18,24,23,27)(20,26,31,28)(21,29,32,25);(1,3,5,2)(4,7,8,6)(9,12,13,10)(11,16,15,14)(17,20,22,31)(18,25,23,29)(19,28,30,26)(21,24,32,27);(1,4,5,8)(2,6,3,7)(17,21,22,32)(18,26,23,28)(19,29,30,25)(20,24,31,27)Polycyclic presentation:
2^8|a^2=h,b^2=fg,c^2=f,d^2=g,e^2=fh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #6698382, #6698387, #6698447, … |
| maximal normal subgroup of | #6698382, #6698387, #6698447, … |
| maximal quotient of | #544618, #545150, #545614, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35721);