Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #2668 |
| automorphism | Aut(G) order 65536 |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C27 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center | C22×C4 | C24 |
| commutator, Frattini, socle, cosocle | C23 | C25 |
| LCS | G ▷ C23 ▷ C1 | |
| UCS | C1 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C23 ▷ C1 | |
1111 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)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18)(19,23)(20,22)(21,24);(1,3)(2,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15)(17,19)(18,21)(20,24)(22,23);(10,14)(13,15)(19,24)(21,23);(2,7)(3,8)(11,16)(13,15)(18,22)(19,24);(1,4,5,6)(2,3,7,8)Polycyclic presentation:
2^8|e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,c^b=ch,d^b=df64 conjugacy classes — view the full character table.
| maximal subgroup of | #131256, #484917, #485485, … |
| maximal normal subgroup of | #131256, #484917, #485485, … |
| maximal quotient of | #520947, #520948, #520949, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 45280);