Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #63011 |
| 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 | |
3313 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 6 generators on 32 points:
(2,7)(3,9)(10,15)(11,16)(17,18,22,25)(19,26,27,23)(20,21,24,30)(28,32,31,29);(1,2)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(19,27)(23,26)(28,31)(29,32);(1,2)(3,8)(4,9)(5,10)(6,7)(11,13)(12,16)(14,15)(17,19)(18,23)(20,28)(21,29)(22,27)(24,31)(25,26)(30,32);(1,3,6,9)(2,8,7,4)(5,11,14,16)(10,13,15,12);(1,4,6,8)(2,9,7,3)(5,13,14,12)(10,11,15,16)(17,20)(18,21)(19,28)(22,24)(23,29)(25,30)(26,32)(27,31);(1,5)(2,10)(3,11)(4,12)(6,14)(7,15)(8,13)(9,16)(17,21)(18,24)(19,29)(20,25)(22,30)(23,31)(26,28)(27,32)Polycyclic presentation:
2^8|a^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,f^e=fh76 conjugacy classes — view the full character table.
| maximal subgroup of | #6350201, #6350202, #6350225, … |
| maximal normal subgroup of | #6350201, #6350202, #6350225, … |
| maximal quotient of | #7627219, #7627230, #7627231, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 56017);