Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63041 |
| 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 | |
3714 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 24 points:
(1,2)(3,7)(4,5)(6,8)(9,10)(11,18)(12,21)(13,19)(14,15)(16,22)(17,23)(20,24);(2,5)(6,7)(10,15)(12,23)(16,18)(19,20);(1,2)(3,6)(4,5)(7,8)(9,11)(10,16)(12,20)(13,21)(14,22)(15,18)(17,24)(19,23);(1,3)(2,6)(4,8)(5,7)(9,12)(10,17)(11,20)(13,18)(14,23)(15,21)(16,24)(19,22);(9,11)(10,18)(12,20)(13,17)(14,22)(15,16)(19,23)(21,24);(9,13)(10,19)(11,21)(12,18)(14,24)(15,20)(16,23)(17,22)Polycyclic presentation:
2^8|b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh70 conjugacy classes — view the full character table.
| maximal subgroup of | #493639, #493648, #511827, … |
| maximal normal subgroup of | #493639, #493648, #511827, … |
| maximal quotient of | #7627401, #7627403, #7627424, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56029);