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, cosocle | C23 | C25 |
| prime radical | C24 | C24 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
716 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,6)(3,4,8,7)(9,10,13,11)(12,15,16,14)(17,18,22,25)(19,23,26,21)(20,24,30,32)(27,31,28,29);(2,6)(4,7)(9,11,13,10)(12,15,16,14)(19,26)(21,23)(27,28)(29,31);(2,6)(3,8)(9,12)(10,14)(11,15)(13,16)(17,19)(18,23)(20,28)(21,25)(22,26)(24,29)(27,30)(31,32);(1,3,5,8)(2,7,6,4)(17,20)(18,24)(19,27)(21,29)(22,30)(23,31)(25,32)(26,28);(1,4,5,7)(2,8,6,3)(17,21,22,23)(18,19,25,26)(20,29,30,31)(24,27,32,28)Polycyclic presentation:
2^8|a^2=gh,b^2=fg,d^2=f,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 | #6679247, #6679250, #6679261, … |
| maximal normal subgroup of | #6679247, #6679250, #6679261, … |
| maximal quotient of | #544452, #545030, #545525, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 35655);