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 | |
1557 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)(9,10,14,18)(11,15,19,21)(12,16,22,24)(13,20,23,17);(2,4)(11,19)(13,23)(15,21)(17,20);(5,6,7,8)(9,11)(10,15)(12,17)(13,16)(14,19)(18,21)(20,22)(23,24);(1,2)(3,4)(9,12)(10,16)(11,20)(13,21)(14,22)(15,23)(17,19)(18,24);(9,13,14,23)(10,17,18,20)(11,16,19,24)(12,21,22,15)Polycyclic presentation:
2^8|a^2=h,c^2=g,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 | #6996497, #6996501, #6996506, … |
| maximal normal subgroup of | #6996497, #6996501, #6996506, … |
| maximal quotient of | #535335, #535380, #535970, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 38122);