Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| automorphism | ? |
| homology | Abelianization (H1) = C26 · Schur multiplier (H2) = C213 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, commutator, Frattini, socle, cosocle | C22 | C26 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ G | |
| LECS | G ▷ C22 ▷ C1 | |
3632 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10,14,18)(11,13,17,15)(12,20,22,16)(19,23,24,21);(2,5)(6,7)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24);(1,2)(3,6)(4,5)(7,8)(9,11)(10,15)(12,19)(13,18)(14,17)(16,23)(20,21)(22,24);(1,3,4,8)(2,6,5,7)(9,12,14,22)(10,16,18,20)(11,19,17,24)(13,21,15,23);(9,11)(10,13)(12,19)(14,17)(15,18)(16,21)(20,23)(22,24);(9,13)(10,17)(11,18)(12,21)(14,15)(16,24)(19,20)(22,23)Polycyclic presentation:
2^8|a^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #346945, #347203, #347207, … |
| maximal normal subgroup of | #346945, #347203, #347207, … |
| maximal quotient of | #7627214, #7627250, #7627260, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56012);