Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #45438 |
| automorphism | Aut(G) order 131072 |
| 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 | |
690 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)(4,7,8,6)(11,15)(13,16)(17,18)(19,25)(20,28)(21,22)(23,27)(24,29)(26,30)(31,32);(2,3)(6,7)(9,10,12,14)(11,13,15,16)(18,22)(20,29)(23,25)(26,32);(9,11,12,15)(10,13,14,16)(17,18)(19,23)(20,24)(21,22)(25,27)(26,30)(28,29)(31,32);(1,3,5,2)(4,7,8,6)(17,19)(18,23)(20,26)(21,27)(22,25)(24,30)(28,31)(29,32);(1,4,5,8)(2,6,3,7)(17,20,21,29)(18,24,22,28)(19,26,27,32)(23,30,25,31)Polycyclic presentation:
2^8|a^2=fh,b^2=g,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh52 conjugacy classes — view the full character table.
| maximal subgroup of | #97193, #97338, #172263, … |
| maximal normal subgroup of | #97193, #97338, #172263, … |
| maximal quotient of | #557080, #557112, #557366, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 39145);