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 | |
1569 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)(9,10,14,17)(11,13,18,15)(12,16,22,24)(19,21,20,23);(1,2,3,4)(11,18)(13,15)(19,20)(21,23);(5,6)(7,8)(9,11)(10,15)(12,20)(13,17)(14,18)(16,21)(19,22)(23,24);(1,2)(3,4)(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20);(9,13,14,15)(10,11,17,18)(12,21,22,23)(16,19,24,20)Polycyclic presentation:
2^8|a^2=h,b^2=f,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 | #6993951, #6993959, #6993969, … |
| maximal normal subgroup of | #6993951, #6993959, #6993969, … |
| maximal quotient of | #535285, #535286, #535922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38081);