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) = C29 |
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 | |
1555 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2)(3,4)(6,8)(12,20)(16,21)(18,24)(22,23);(1,2,3,4)(10,14)(12,20)(15,17)(18,24);(5,6)(7,8)(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22);(1,2)(3,4)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23);(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^8|b^2=f,c^2=h,d^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 | #132977, #487043, #487171, … |
| maximal normal subgroup of | #132977, #487043, #487171, … |
| maximal quotient of | #535284, #535918, #536485, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37748);