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 | |
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:
(1,2)(3,10)(4,5)(6,7)(8,13)(9,14)(11,16)(12,15)(18,22)(20,27)(23,28)(26,31);(2,7)(4,9)(8,10)(11,15)(19,25)(23,28)(26,31)(30,32);(1,2,6,7)(3,8,13,10)(4,5,9,14)(11,12,15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(1,3,6,13)(2,8,7,10)(4,11,9,15)(5,12,14,16);(1,4,6,9)(2,5,7,14)(3,11,13,15)(8,12,10,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,5)(2,9)(3,12)(4,7)(6,14)(8,15)(10,11)(13,16)(17,20)(18,24)(19,26)(21,27)(22,29)(23,30)(25,31)(28,32)Polycyclic presentation:
2^8|c^2=h,d^2=g,e^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 | #186157, #186164, #186171, … |
| maximal normal subgroup of | #186157, #186164, #186171, … |
| maximal quotient of | #7626214, #7626223, #7626225, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55815);