Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18810 |
| 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 | #3499 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
540 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 540 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 128 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2,5,8)(3,7,4,6)(9,10)(11,15)(12,22)(13,25)(14,17)(16,24)(18,28)(19,35)(20,33)(21,30)(23,34)(26,31)(27,32)(29,37)(36,39)(38,40);(2,6)(3,4)(7,8)(9,11,14,21)(10,15,17,30)(12,19,23,36)(13,26,28,20)(16,29,32,40)(18,33,25,31)(22,39,34,35)(24,38,27,37);(9,12)(10,16)(11,19)(13,27)(14,23)(15,29)(17,32)(18,34)(20,38)(21,36)(22,25)(24,28)(26,37)(30,40)(31,39)(33,35);(1,3,5,4)(2,7,8,6)(10,17)(15,30)(16,32)(18,25)(22,34)(29,40)(31,33)(35,39)Polycyclic presentation:
2^8|a^2=g,b^2=h,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh37 conjugacy classes — view the full character table.
| maximal subgroup of | #249801, #249803, #249815, … |
| maximal normal subgroup of | #249801, #249803, #249815, … |
| maximal quotient of | #168926, #168981, #169026, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11815);