Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) order 16384 |
| 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 | C24 | C22×C4 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
813 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)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,21,25,22,26,32,38)(19,23,29,35,28,34,41,45)(20,24,31,37,30,36,42,46)(27,33,40,44,39,43,47,48);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19)(18,23)(20,27)(21,29)(22,28)(24,33)(25,35)(26,34)(30,39)(31,40)(32,41)(36,43)(37,44)(38,45)(42,47)(46,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(17,20)(18,24)(19,27)(21,31)(22,30)(23,33)(25,37)(26,36)(28,39)(29,40)(32,42)(34,43)(35,44)(38,46)(41,47)(45,48);(2,8)(6,9)(7,16)(14,15)(20,30)(24,36)(27,39)(31,42)(33,43)(37,46)(40,47)(44,48);(19,28)(23,34)(27,39)(29,41)(33,43)(35,45)(40,47)(44,48)Polycyclic presentation:
2^8|a^2=g,b^2=f,c^2=f,g^2=h,b^a=bf,d^a=df,c^b=cf,e^b=eh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | #129696, #209564, #209565, … |
| maximal normal subgroup of | #129696, #209564, #209565, … |
| maximal quotient of | #6370201, #6370223, #6370327, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53240);