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) = C26×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C23 | C25 |
| prime radical | #2733 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
862 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(2,4)(3,7)(9,10,15,13)(11,16,14,12)(17,18,21,20)(19,23,24,22);(1,2,5,4)(3,8,7,6)(9,11,15,14)(10,16,13,12);(1,2,5,4)(3,6,7,8)(9,12)(10,14)(11,13)(15,16)(17,19,21,24)(18,22,20,23);(1,3,5,7)(2,6,4,8)(9,13,15,10)(11,16,14,12);(1,4,5,2)(3,8,7,6)(9,14,15,11)(10,16,13,12)(17,20,21,18)(19,23,24,22)Polycyclic presentation:
2^8|a^2=fg,b^2=fh,c^2=gh,d^2=fh,e^2=g,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 | #97285, #129631, #131138, … |
| maximal normal subgroup of | #97285, #129631, #131138, … |
| maximal quotient of | #553941, #554148, #554622, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 30255);