Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #60756 |
| 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 |
| prime radical | #3494 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×#117 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
871 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18)(19,26)(20,24)(21,23)(22,27)(25,29)(28,30)(31,32);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,20,27)(18,22,24,26)(21,28,29,32)(23,31,25,30);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,23)(19,27)(21,29)(22,30)(24,25)(26,31);(2,8)(6,9)(7,16)(14,15)(18,24)(22,26)(23,25)(30,31);(19,27)(22,26)(28,32)(30,31)Polycyclic presentation:
2^8|b^2=f,c^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #249978, #249982, #249986, … |
| maximal normal subgroup of | #249978, #249982, #249986, … |
| maximal quotient of | #6374370, #6374544, #6375009, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53861);