Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19050 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #532 | C22 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
403 conjugacy classes of subgroups; only the 111 normal subgroups are shown.
This group has 403 conjugacy classes of subgroups — too many to draw. Only its 111 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 48 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 48 points:
(3,9)(4,11)(6,14)(7,12)(10,13)(15,16)(17,18)(19,29)(20,34)(21,26)(22,27)(23,28)(24,32)(25,36)(30,38)(31,39)(33,42)(35,40)(37,41)(43,45)(44,47)(46,48);(1,2,5,8)(3,6,10,15)(4,12,11,7)(9,16,13,14)(17,19)(18,24)(20,30)(21,31)(22,32)(23,33)(25,38)(26,29)(27,42)(28,39)(34,47)(35,44)(36,45)(37,46)(40,48)(41,43);(1,3)(2,6)(4,13)(5,10)(7,16)(8,15)(9,11)(12,14)(17,20,23,37)(18,25,28,40)(19,30,33,46)(21,36,22,35)(24,38,39,48)(26,41,27,34)(29,43,42,47)(31,45,32,44);(17,21,23,22)(18,26,28,27)(19,31,33,32)(20,35,37,36)(24,29,39,42)(25,34,40,41)(30,44,46,45)(38,47,48,43)Polycyclic presentation:
2^8|b^2=h,c^2=g,d^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,e^a=eg,f^a=fgh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fgh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153486, #153492, #245309, … |
| maximal normal subgroup of | #153486, #153492, #245309, … |
| maximal quotient of | #167777, #167812, #167828, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12051);