Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #62892 |
| 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 | |
3314 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 6 generators on 32 points:
(1,2,5,10)(3,11,12,7)(4,15,17,9)(6,8,18,23)(13,20,25,24)(14,19,26,21)(16,28,29,22)(27,31,32,30);(1,3,6,14)(2,7,8,21)(4,13,16,27)(5,12,18,26)(9,20,22,31)(10,11,23,19)(15,24,28,30)(17,25,29,32);(2,8)(3,12)(7,19)(9,22)(10,23)(11,21)(13,25)(14,26)(15,28)(20,30)(24,31)(27,32);(1,4,5,17)(2,9,10,15)(3,13,12,25)(6,16,18,29)(7,20,11,24)(8,22,23,28)(14,27,26,32)(19,30,21,31);(4,16)(9,22)(13,27)(15,28)(17,29)(20,31)(24,30)(25,32);(2,8)(7,21)(9,22)(10,23)(11,19)(15,28)(20,31)(24,30)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh73 conjugacy classes — view the full character table.
| Fitting subgroup of | #62903⋊C3 |
| maximal subgroup of | #6352588, #6352618, #6352648, … |
| maximal normal subgroup of | #6352588, #6352618, #6352648, … |
| maximal quotient of | #7626581, #7626665, #7626677, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55891);