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 | #3561 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
828 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 64 |
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)(17,18,21,25)(19,27,29,23)(20,31,32,47)(22,35,34,26)(24,38,39,33)(28,42,43,46)(30,37,45,41)(36,44,40,48);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,10)(14,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,36)(25,27)(26,37)(31,46)(32,43)(33,44)(34,45)(35,41)(38,48)(39,40)(42,47);(1,4,6,5)(2,8,9,14)(3,11,12,13)(7,15,10,16)(17,20,22,33,21,32,34,38)(18,24,26,31,25,39,35,47)(19,28,30,44,29,43,45,48)(23,36,37,46,27,40,41,42);(3,12)(7,10)(11,13)(15,16)(19,29)(20,32)(23,27)(24,39)(30,45)(31,47)(33,38)(37,41);(1,5,6,4)(2,8,9,14)(3,13,12,11)(7,15,10,16)(18,25)(19,29)(24,39)(26,35)(28,43)(30,45)(31,47)(44,48)Polycyclic presentation:
2^8|a^2=h,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 | #250474, #250478, #250482, … |
| maximal normal subgroup of | #250474, #250478, #250482, … |
| maximal quotient of | #6386959, #6389657, #6389683, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55559);