Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| automorphism | Aut(G) order 131072 |
| 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 | |
754 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 32 points:
(3,7)(6,8)(11,15)(12,14)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31);(2,5)(6,8)(9,10,13,16)(11,12,15,14)(17,18)(19,25)(20,23)(21,24)(22,27)(26,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,14,16,12);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19,21,27)(18,22,24,25)(20,26,29,32)(23,30,28,31);(1,3,4,7)(2,6,5,8)(9,12,13,14)(10,15,16,11)(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31)Polycyclic presentation:
2^8|a^2=f,b^2=g,c^2=g,d^2=f,e^2=fh,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^b=dfh52 conjugacy classes — view the full character table.
| maximal subgroup of | #133250, #133251, #251740, … |
| maximal normal subgroup of | #133250, #133251, #251740, … |
| maximal quotient of | #553764, #553796, #553997, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30308);