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 | #3515 | 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)(17,18,22,27)(19,28,31,24)(20,33,35,40)(21,26,23,36)(25,37,41,34)(29,44,46,48)(30,43,32,39)(38,45,42,47);(1,3)(2,6)(4,11)(5,10)(7,14)(8,9)(12,15)(13,16)(17,19,22,31)(18,24,27,28)(20,29,35,46)(21,30,23,32)(25,38,41,42)(26,39,36,43)(33,48,40,44)(34,45,37,47);(17,20)(18,25)(19,29)(21,34)(22,35)(23,37)(24,38)(26,40)(27,41)(28,42)(30,45)(31,46)(32,47)(33,36)(39,44)(43,48);(3,10)(6,9)(11,16)(14,15)(17,21,22,23)(18,26,27,36)(19,30,31,32)(20,34,35,37)(24,39,28,43)(25,40,41,33)(29,45,46,47)(38,44,42,48);(1,4,5,13)(2,7,8,12)(3,11,10,16)(6,14,9,15)(18,27)(19,31)(25,41)(26,36)(29,46)(30,32)(33,40)(45,47)Polycyclic presentation:
2^8|a^2=f,b^2=h,d^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 | #6364557, #6364561, #6364589, … |
| maximal normal subgroup of | #6364557, #6364561, #6364589, … |
| maximal quotient of | #6386459, #6388579, #6388587, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 55205);