Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62817 |
| 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 | |
3318 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)(3,9)(4,10)(5,13)(6,7)(8,14)(11,12)(15,16)(18,22)(20,27)(23,28)(26,31);(2,7)(3,10)(8,13)(11,16)(19,25)(23,28)(26,31)(30,32);(1,3)(2,4)(5,11)(6,10)(7,9)(8,12)(13,15)(14,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,4)(2,3)(5,12)(6,9)(7,10)(8,11)(13,16)(14,15);(1,2)(3,4)(5,8)(6,7)(9,10)(11,12)(13,14)(15,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,5,6,14)(2,8,7,13)(3,11,10,16)(4,12,9,15)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|c^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^b=eh76 conjugacy classes — view the full character table.
| maximal subgroup of | #185316, #185318, #185320, … |
| maximal normal subgroup of | #185316, #185318, #185320, … |
| maximal quotient of | #7626216, #7626228, #7626252, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55817);