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 | |
720 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(11,15)(14,16)(17,18)(19,26)(20,21)(22,23)(24,29)(25,30)(27,31)(28,32);(1,2,4,6)(3,7,8,5)(9,10,13,12)(11,14,15,16)(18,23)(21,25)(24,26)(28,31);(1,2)(3,5)(4,6)(7,8)(9,11)(10,14)(12,16)(13,15)(17,19,22,29)(18,24,23,26)(20,27,30,32)(21,28,25,31);(9,12,13,10)(11,16,15,14)(17,20)(18,25)(19,27)(21,23)(22,30)(24,31)(26,28)(29,32);(1,3,4,8)(2,5,6,7)(17,21,22,25)(18,20,23,30)(19,28,29,31)(24,27,26,32)Polycyclic presentation:
2^8|b^2=fgh,c^2=f,d^2=fg,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 | #6702461, #6702464, #6702524, … |
| maximal normal subgroup of | #6702461, #6702464, #6702524, … |
| maximal quotient of | #521638, #522272, #522946, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32238);