Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #42996 |
| 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 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
956 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)(11,15)(13,16)(17,18,21,22)(19,23,24,20);(1,3,5,8)(2,6,4,7)(9,10)(11,16)(12,14)(13,15)(18,22)(20,23);(3,8)(6,7)(9,10,12,14)(11,13,15,16)(17,19)(18,20)(21,24)(22,23);(1,3)(2,7)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,23)(22,24);(1,4,5,2)(3,6,8,7)(9,11,12,15)(10,13,14,16)Polycyclic presentation:
2^8|a^2=g,b^2=fg,c^2=fgh,e^2=h,b^a=bf,c^a=cg,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh58 conjugacy classes — view the full character table.
| maximal subgroup of | #7005687, #7005707, #7005762, … |
| maximal normal subgroup of | #7005687, #7005707, #7005762, … |
| maximal quotient of | #535450, #535624, #536053, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38361);