Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18284 |
| 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 | C22×C4 | C24 |
| prime radical | #3455 | C2 |
| LCS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C22 ▷ C1 | |
465 conjugacy classes of subgroups; only the 119 normal subgroups are shown.
This group has 465 conjugacy classes of subgroups — too many to draw. Only its 119 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 40 points:
(2,6)(3,4)(7,8)(9,10,14,18)(11,15,21,28)(12,16,23,30)(13,24,25,17)(19,31,33,26)(20,34,35,27)(22,36,37,29)(32,38,40,39);(1,2)(3,6)(4,8)(5,7)(9,11)(10,15)(12,19)(13,20)(14,21)(16,26)(17,27)(18,28)(22,32)(23,33)(24,34)(25,35)(29,38)(30,31)(36,39)(37,40);(1,3,5,4)(2,6,7,8)(9,12)(10,16)(11,19)(13,22)(14,23)(15,26)(17,29)(18,30)(20,32)(21,33)(24,36)(25,37)(27,38)(28,31)(34,39)(35,40);(9,13,14,25)(10,17,18,24)(11,20,21,35)(12,22,23,37)(15,27,28,34)(16,29,30,36)(19,32,33,40)(26,38,31,39)Polycyclic presentation:
2^8|a^2=h,c^2=g,d^2=h,e^2=g,b^a=be,c^a=cf,d^a=dh,e^a=eg,e^b=eg,f^b=fh,e^c=eh40 conjugacy classes — view the full character table.
| maximal subgroup of | #247705, #247706, #247709, … |
| maximal normal subgroup of | #247705, #247706, #247709, … |
| maximal quotient of | #168552, #168554, #168609, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11300);