Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18930 |
| automorphism | Aut(G) order 8192 |
| 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 | #3554 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
437 conjugacy classes of subgroups; only the 113 normal subgroups are shown.
This group has 437 conjugacy classes of subgroups — too many to draw. Only its 113 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:
(1,2)(3,7)(4,6)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,23,28)(19,24,31,39)(20,32,34,42)(21,27,22,26)(25,36,40,33)(29,43,45,47)(30,41,35,38)(37,44,48,46);(2,6)(4,12)(7,13)(8,9)(11,16)(14,15)(17,19)(18,24)(20,29)(21,30)(22,35)(23,31)(25,37)(26,38)(27,41)(28,39)(32,47)(33,44)(34,45)(36,46)(40,48)(42,43);(1,3,5,10)(2,7,9,14)(4,11,12,16)(6,13,8,15)(17,20)(18,25)(19,29)(21,33)(22,36)(23,34)(24,37)(26,32)(27,42)(28,40)(30,44)(31,45)(35,46)(38,47)(39,48)(41,43);(3,10)(7,14)(11,16)(13,15)(17,21,23,22)(18,26,28,27)(19,30,31,35)(20,33,34,36)(24,38,39,41)(25,32,40,42)(29,44,45,46)(37,47,48,43)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,e^b=eg,f^b=fh,d^c=dg,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #347426, #347427, #347430, … |
| maximal normal subgroup of | #347426, #347427, #347430, … |
| maximal quotient of | #170421, #170457, #170517, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11958);