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 | |
3248 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)(3,8)(4,7)(5,9)(6,11)(10,15)(12,13)(14,16)(17,18)(19,26)(20,21)(22,24)(23,27)(25,32)(28,29)(30,31);(1,2)(3,8)(4,12)(5,9)(6,11)(7,13)(10,14)(15,16)(17,19,23,22)(18,24,27,26)(20,30,25,28)(21,29,32,31);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,15)(13,16)(17,20,23,25)(18,21,27,32)(19,28,22,30)(24,31,26,29);(1,4)(2,7)(3,10)(5,13)(6,14)(8,15)(9,12)(11,16)(17,21)(18,25)(19,29)(20,27)(22,31)(23,32)(24,28)(26,30);(1,3)(2,8)(4,10)(5,11)(6,9)(7,15)(12,14)(13,16)(17,19,23,22)(18,26,27,24)(20,28,25,30)(21,29,32,31);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,13)(15,16)(17,22,23,19)(18,24,27,26)(20,30,25,28)(21,31,32,29)Polycyclic presentation:
2^8|b^2=g,c^2=g,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 | #347446, #347466, #347492, … |
| maximal normal subgroup of | #347446, #347466, #347492, … |
| maximal quotient of | #7627412, #7627413, #7627448, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56044);