Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #19086 |
| automorphism | Aut(G) = C2×#72286730 (order 4096) · Inn(G) = C2×#120 · Out(G) = C2×#120 |
| 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 | #3561 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
427 conjugacy classes of subgroups; only the 109 normal subgroups are shown.
This group has 427 conjugacy classes of subgroups — too many to draw. Only its 109 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) = 64 |
A minimal faithful permutation representation — 4 generators on 48 points:
(1,2)(3,9)(4,12)(5,7)(6,11)(8,13)(10,16)(14,15)(17,18,23,28)(19,29,32,39)(20,33,35,41)(21,26,22,27)(24,36,40,31)(25,37,42,34)(30,38,46,48)(43,44,47,45);(4,13)(8,12)(9,16)(11,14)(17,19,23,32)(18,24,28,40)(20,30,35,46)(21,36,22,31)(25,38,42,48)(26,29,27,39)(33,47,41,43)(34,45,37,44);(1,3)(2,6)(4,14)(5,10)(7,15)(8,16)(9,12)(11,13)(17,20)(18,25)(19,30)(21,34)(22,37)(23,35)(24,38)(26,41)(27,33)(28,42)(29,43)(31,44)(32,46)(36,45)(39,47)(40,48);(2,7)(6,15)(8,12)(9,16)(17,21,23,22)(18,26,28,27)(19,31,32,36)(20,34,35,37)(24,39,40,29)(25,41,42,33)(30,44,46,45)(38,47,48,43)Polycyclic presentation:
2^8|a^2=g,b^2=g,d^2=g,e^2=g,f^2=gh,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fgh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fgh31 conjugacy classes — view the full character table.
| maximal subgroup of | #250456, #250461, #250464, … |
| maximal normal subgroup of | #250456, #250461, #250464, … |
| maximal quotient of | #168916, #168918, #168922, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 12079);