Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #36702 |
| 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 | C2×#121 | C22 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
836 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)(3,5)(4,7)(6,8)(11,15)(14,16)(17,18,21,19)(20,23,24,22);(2,3)(6,7)(10,13)(14,16)(17,19,21,18)(20,23,24,22);(9,10)(11,14)(12,13)(15,16)(17,20,21,24)(18,22,19,23);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16);(1,4,5,8)(2,6,3,7)(9,11,12,15)(10,14,13,16)Polycyclic presentation:
2^8|a^2=g,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 | #6492957, #6492961, #6492967, … |
| maximal normal subgroup of | #6492957, #6492961, #6492967, … |
| maximal quotient of | #553770, #553790, #553992, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30315);