Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | ? |
| homology | Abelianization (H1) = C24×C4 · Schur multiplier (H2) = C28 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, Frattini, cosocle | C2×C4 | C25 |
| commutator, socle | C22 | C24×C4 |
| prime radical | #3531 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
814 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)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,26,22,24,33,37)(19,23,29,36,27,34,41,45)(20,30,31,44,32,25,38,39)(28,42,43,48,40,35,46,47);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,23)(20,28)(21,29)(22,27)(24,34)(25,35)(26,36)(30,42)(31,43)(32,40)(33,41)(37,45)(38,46)(39,47)(44,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(18,24)(19,27)(25,30)(26,37)(28,40)(29,41)(39,44)(43,46);(2,8)(6,9)(7,16)(14,15)(19,27)(23,34)(28,40)(29,41)(35,42)(36,45)(43,46)(47,48);(17,20,21,31,22,32,33,38)(18,25,26,39,24,30,37,44)(19,28,29,43,27,40,41,46)(23,35,36,47,34,42,45,48)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,e^2=g,g^2=h,b^a=bf,c^a=ch,d^a=df,e^a=eh,c^b=cfh,d^b=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129671, #210394, #210395, … |
| maximal normal subgroup of | #129671, #210394, #210395, … |
| maximal quotient of | #6370574, #6370579, #6370615, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53326);