Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | |
770 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(9,10,13,16)(11,12,15,14)(17,18)(19,22)(20,24)(21,29)(23,26)(25,31)(27,32)(28,30);(2,6)(4,7)(11,15)(12,14)(17,19,23,20)(18,24,26,22)(21,27,31,28)(25,32,29,30);(1,2)(3,7)(4,8)(5,6)(9,11,13,15)(10,14,16,12)(17,20,23,19)(18,24,26,22)(21,28,31,27)(25,32,29,30);(1,3,5,8)(2,7,6,4)(17,21,23,31)(18,25,26,29)(19,27,20,28)(22,30,24,32);(1,4,5,7)(2,8,6,3)(9,12,13,14)(10,15,16,11)(17,22,23,24)(18,19,26,20)(21,30,31,32)(25,27,29,28)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=fg,d^2=fgh,e^2=fh,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 | #6702454, #6702459, #6702519, … |
| maximal normal subgroup of | #6702454, #6702459, #6702519, … |
| maximal quotient of | #521987, #522522, #523084, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 33166);