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 | |
3360 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)(19,24)(20,23)(21,25)(22,27)(26,31)(28,29)(30,32);(2,8)(4,14)(9,11)(12,15)(17,18)(19,24)(20,28)(21,25)(22,27)(23,29)(26,30)(31,32);(1,3,7,5)(2,9,8,11)(4,12,14,15)(6,13,10,16)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(1,4)(2,10)(3,12)(5,15)(6,8)(7,14)(9,16)(11,13)(17,20)(18,23)(19,26)(21,29)(22,30)(24,31)(25,28)(27,32);(1,5,7,3)(2,9,8,11)(4,15,14,12)(6,13,10,16)(17,19)(18,24)(20,26)(21,27)(22,25)(23,31)(28,30)(29,32);(1,6,7,10)(2,4,8,14)(3,13,5,16)(9,12,11,15)(17,18)(19,22)(20,23)(21,25)(24,27)(26,30)(28,29)(31,32)Polycyclic presentation:
2^8|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 | #347442, #347462, #347484, … |
| maximal normal subgroup of | #347442, #347462, #347484, … |
| maximal quotient of | #7627408, #7627409, #7627437, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56039);