Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #61629 |
| 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 | #3516 | C2 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ #529 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
821 conjugacy classes of subgroups.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 5 generators on 48 points:
(1,2,5,8)(3,9,10,6)(4,12,13,7)(11,14,16,15)(17,18)(19,27)(20,31)(21,25)(22,35)(23,29)(24,33)(26,34)(28,43)(30,38)(32,40)(36,39)(37,47)(41,45)(42,46)(44,48);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19)(18,23)(20,28)(21,29)(22,30)(24,37)(25,27)(26,38)(31,48)(32,44)(33,45)(34,46)(35,42)(36,47)(39,41)(40,43);(17,20,21,32)(18,24,25,39)(19,28,29,44)(22,36,34,33)(23,37,27,41)(26,40,35,31)(30,47,46,45)(38,43,42,48);(3,10)(6,9)(11,16)(14,15);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,25)(19,29)(24,39)(26,35)(28,44)(30,46)(31,40)(45,47)Polycyclic presentation:
2^8|a^2=fh,c^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=dfh,e^b=eh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#62214, #10579475, #10579512, … |
| maximal normal subgroup of | C2×#62214, #10579475, #10579512, … |
| maximal quotient of | #6386457, #6388573, #6388591, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55202);