Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| automorphism | Aut(G) order 262144 |
| 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 | |
707 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,6,3)(4,7,5,8)(11,15)(13,16)(17,18)(19,21)(20,28)(22,23)(24,26)(25,30)(27,31)(29,32);(1,3,6,2)(4,7,5,8)(9,10,12,14)(11,13,15,16)(18,23)(19,26)(25,28)(27,32);(1,4,6,5)(2,7,3,8)(9,11,12,15)(10,13,14,16)(17,19)(18,24)(20,27)(21,23)(22,26)(25,31)(28,29)(30,32);(17,20,22,30)(18,25,23,28)(19,27,26,32)(21,29,24,31);(1,5,6,4)(2,8,3,7)(17,21,22,24)(18,19,23,26)(20,29,30,31)(25,27,28,32)Polycyclic presentation:
2^8|a^2=fh,b^2=gh,c^2=gh,d^2=f,e^2=h,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh58 conjugacy classes — view the full character table.
| maximal subgroup of | #98017, #130675, #166443, … |
| maximal normal subgroup of | #98017, #130675, #166443, … |
| maximal quotient of | #521499, #522151, #522657, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32745);