Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
971 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(10,11)(14,15)(18,21)(22,24);(2,5)(6,7)(9,10,13,11)(12,15,16,14)(18,21)(19,23);(9,10,13,11)(12,14,16,15)(17,18,20,21)(19,22,23,24);(1,3,4,8)(2,6,5,7)(9,11,13,10)(12,15,16,14);(1,2)(3,6)(4,5)(7,8)(9,12)(10,14)(11,15)(13,16)(17,19,20,23)(18,22,21,24)Polycyclic presentation:
2^8|b^2=gh,c^2=g,d^2=f,e^2=h,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 | #130572, #130898, #254184, … |
| maximal normal subgroup of | #130572, #130898, #254184, … |
| maximal quotient of | #521627, #522261, #522915, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32173);