Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #62283 |
| automorphism | ? |
| homology | Abelianization (H1) = C25 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C23×D4 |
| commutator, Frattini, cosocle | C2×C4 | C25 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
888 conjugacy classes of subgroups.
| minimal degree | μ(G) = 32 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 5 generators on 32 points:
(1,2)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,26)(20,23)(21,25)(22,29)(24,27)(28,31)(30,32);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,18)(19,22)(20,24)(21,25)(23,27)(26,29)(28,30)(31,32);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,19,27,32,21,28,20,29)(18,22,23,31,25,30,24,26);(3,12)(7,10)(11,13)(15,16)(19,28)(22,30)(26,31)(29,32);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(17,20,21,27)(18,23,25,24)(19,29,28,32)(22,31,30,26)Polycyclic presentation:
2^8|c^2=fgh,e^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,d^c=dh46 conjugacy classes — view the full character table.
| maximal subgroup of | #250473, #250477, #250481, … |
| maximal normal subgroup of | #250473, #250477, #250481, … |
| maximal quotient of | #6386959, #6389656, #6389682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55558);