Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #37644 |
| 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 | #2537 | C2 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
874 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,6,5)(3,8,4,7)(10,13)(14,16);(2,5)(7,8)(17,18,20,22)(19,23,24,21);(1,3,6,4)(2,7,5,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,21)(20,22)(23,24);(1,4,6,3)(2,8,5,7);(1,5,6,2)(3,8,4,7)(9,11,12,15)(10,14,13,16)(17,19,20,24)(18,21,22,23)Polycyclic presentation:
2^8|a^2=f,b^2=h,c^2=f,d^2=f,e^2=gh,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 | #6545985, #6545988, #6545997, … |
| maximal normal subgroup of | #6545985, #6545988, #6545997, … |
| maximal quotient of | #521616, #522250, #522895, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 32249);