Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| 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 | |
3593 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 24 points:
(2,5)(6,8)(9,10,16,19)(11,20,14,17)(12,15,13,18)(21,22,23,24);(9,11,16,14)(10,17,19,20)(12,23,13,21)(15,22,18,24);(1,2)(3,6)(4,5)(7,8)(9,12,16,13)(10,18,19,15)(11,21,14,23)(17,22,20,24);(3,7)(6,8)(9,13,16,12)(10,15,19,18)(11,21,14,23)(17,22,20,24);(1,3,4,7)(2,6,5,8)(9,14,16,11)(10,17,19,20)(12,23,13,21)(15,24,18,22);(1,2)(3,6)(4,5)(7,8)(9,15)(10,12)(11,22)(13,19)(14,24)(16,18)(17,23)(20,21)Polycyclic presentation:
2^8|a^2=g,b^2=g,c^2=g,d^2=g,e^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh73 conjugacy classes — view the full character table.
| Fitting subgroup of | #62917⋊C3 |
| maximal subgroup of | #347071, #347095, #347119, … |
| maximal normal subgroup of | #347071, #347095, #347119, … |
| maximal quotient of | #7626596, #7626652, #7626690, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55905);