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 | #537 | C22 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C4×#41 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
831 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2)(3,9)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,22,26)(19,23,28,37)(20,30,32,40)(21,33,35,25)(24,34,39,31)(27,41,42,48)(29,45,44,38)(36,46,47,43);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19,22,28)(18,23,26,37)(20,27,32,42)(21,29,35,44)(24,36,39,47)(25,38,33,45)(30,41,40,48)(31,43,34,46);(3,10)(6,9)(11,16)(13,15)(17,20,22,32)(18,24,26,39)(19,27,28,42)(21,34,35,31)(23,36,37,47)(25,40,33,30)(29,46,44,43)(38,48,45,41);(2,7)(6,9)(8,14)(13,15)(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46)Polycyclic presentation:
2^8|a^2=h,b^2=h,c^2=h,e^2=f,g^2=h,b^a=bf,c^a=cg,d^a=df,g^a=gh,c^b=cf,d^b=dh,e^b=eh,g^c=gh49 conjugacy classes — view the full character table.
| maximal subgroup of | #6350757, #6350759, #6350805, … |
| maximal normal subgroup of | #6350757, #6350759, #6350805, … |
| maximal quotient of | #6374684, #6374705, #6375088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53983);