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 | |
3246 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2)(3,11)(4,6)(5,9)(7,8)(10,14)(12,16)(13,15)(17,18,23,24)(19,20,27,32)(21,25,22,26)(28,30,29,31);(2,8)(4,14)(9,11)(12,15)(18,24)(19,27)(25,26)(28,29);(1,3,7,5)(2,9,8,11)(4,12,14,15)(6,13,10,16)(17,19)(18,20)(21,28)(22,29)(23,27)(24,32)(25,30)(26,31);(1,4)(2,10)(3,12)(5,15)(6,8)(7,14)(9,16)(11,13)(17,20)(18,19)(21,30)(22,31)(23,32)(24,27)(25,28)(26,29);(1,5,7,3)(2,9,8,11)(4,15,14,12)(6,13,10,16)(17,21,23,22)(18,25,24,26)(19,28,27,29)(20,30,32,31);(1,6,7,10)(2,4,8,14)(3,13,5,16)(9,12,11,15)(17,22,23,21)(18,26,24,25)(19,29,27,28)(20,31,32,30)Polycyclic presentation:
2^8|a^2=gh,c^2=h,e^2=g,f^2=g,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 | #347439, #347459, #347480, … |
| maximal normal subgroup of | #347439, #347459, #347480, … |
| maximal quotient of | #7627419, #7627420, #7627496, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56053);