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 |
| prime radical | #3554 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
826 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)(3,10)(4,8)(5,14)(6,9)(7,12)(11,16)(13,15)(17,18)(19,28)(20,32)(21,26)(22,27)(23,37)(24,31)(25,33)(29,44)(30,40)(34,41)(35,39)(36,42)(38,45)(43,47)(46,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,24)(20,29)(21,35)(22,31)(23,30)(25,38)(26,40)(27,28)(32,48)(33,47)(34,46)(36,45)(37,39)(41,44)(42,43);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,21,36,22,34,23,33)(18,25,37,41,27,42,26,32)(19,29,35,45,31,46,30,47)(24,38,39,44,28,43,40,48);(3,12)(7,10)(11,13)(15,16)(17,21,22,23)(18,26,27,37)(19,30,31,35)(20,33,34,36)(24,39,28,40)(25,41,42,32)(29,45,46,47)(38,48,43,44);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,27)(19,31)(25,42)(26,37)(29,46)(30,35)(32,41)(45,47)Polycyclic presentation:
2^8|c^2=fg,d^2=h,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 | #250490, #250494, #250498, … |
| maximal normal subgroup of | #250490, #250494, #250498, … |
| maximal quotient of | #6386958, #6389657, #6389684, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55560);