Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #53158 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
735 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2,5,3)(4,7,8,6)(9,10,14,11)(12,16,13,15)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31);(2,3)(6,7)(10,11)(12,13)(17,19,22,20)(18,23,25,27)(21,26,29,28)(24,30,31,32);(9,11,14,10)(12,16,13,15)(17,20,22,19)(18,23,25,27)(21,28,29,26)(24,30,31,32);(1,3,5,2)(4,7,8,6)(9,12,14,13)(10,15,11,16)(17,21,22,29)(18,24,25,31)(19,26,20,28)(23,30,27,32);(1,4,5,8)(2,6,3,7)(9,13,14,12)(10,16,11,15)(17,18)(19,27)(20,23)(21,24)(22,25)(26,32)(28,30)(29,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=gh,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=df,e^b=egh46 conjugacy classes — view the full character table.
| maximal subgroup of | #98125, #179371, #179372, … |
| maximal normal subgroup of | #98125, #179371, #179372, … |
| maximal quotient of | #606806, #606817, #606835, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 46354);