Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 2 |
| isoclinism | isoclinic class of #3532 |
| automorphism | Aut(G) = #72234530 (order 4096) · Inn(G) = C25 · Out(G) = C2×#379 |
| 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 | #3455 | C2 |
| LCS | G ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C2×C4 ◁ G | |
| LECS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
842 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)(4,8)(5,7)(6,10)(11,15)(12,14)(13,16)(17,18,21,25,22,26,33,40)(19,23,29,36,28,35,42,46)(20,24,32,39,31,38,44,48)(27,34,41,45,30,37,43,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,13)(12,16)(14,15)(17,19)(18,23)(20,30)(21,29)(22,28)(24,37)(25,36)(26,35)(27,31)(32,43)(33,42)(34,38)(39,47)(40,46)(41,44)(45,48);(3,10)(6,9)(11,16)(13,15)(17,20)(18,24)(19,27)(21,32)(22,31)(23,34)(25,39)(26,38)(28,30)(29,41)(33,44)(35,37)(36,45)(40,48)(42,43)(46,47);(2,7)(6,9)(8,14)(13,15)(20,31)(24,38)(27,30)(32,44)(34,37)(39,48)(41,43)(45,47);(1,4,5,12)(2,8,7,14)(3,11,10,16)(6,13,9,15)(19,28)(23,35)(27,30)(29,42)(34,37)(36,46)(41,43)(45,47)Polycyclic presentation:
2^8|a^2=g,e^2=f,g^2=h,b^a=bf,d^a=df,c^b=cfh,e^b=eh,d^c=dh76 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#60297, #10574413, #10574417, … |
| maximal normal subgroup of | C2×#60297, #10574413, #10574417, … |
| maximal quotient of | #6370255, #6370258, #6370272, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 53285);