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 | |
1507 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 24 points:
(1,2,4,3)(6,8)(9,10)(11,14)(12,15)(13,18)(16,21)(17,19)(20,23)(22,24);(2,3)(10,11)(12,19)(13,21)(22,23);(5,6)(7,8)(9,11,14,10)(12,15,19,17)(13,18,21,16)(20,22,24,23);(1,3,4,2)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23);(9,13)(10,16)(11,18)(12,20)(14,21)(15,22)(17,23)(19,24)Polycyclic presentation:
2^8|a^2=fh,c^2=h,d^2=f,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 | #130757, #245555, #245556, … |
| maximal normal subgroup of | #130757, #245555, #245556, … |
| maximal quotient of | #535329, #535378, #535961, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 37435);