Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23 | H(5,𝔽2) |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
321 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 321 conjugacy classes of subgroups — too many to draw. Only its 115 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:
(1,2,5,7)(3,4,8,6)(9,10,13,18,14,19,29,36)(11,15,21,17,23,32,22,31)(12,24,26,34,27,35,28,16)(20,37,38,40,39,33,25,30);(1,3,5,8)(2,6,7,4)(9,11)(10,15)(12,25)(13,21)(14,23)(16,33)(17,18)(19,32)(20,26)(22,29)(24,30)(27,38)(28,39)(31,36)(34,37)(35,40);(9,12,14,27)(10,16,19,34)(11,20,23,39)(13,28,29,26)(15,30,32,40)(17,33,31,37)(18,35,36,24)(21,25,22,38);(1,4,5,6)(2,3,7,8)(9,11)(10,17)(12,20)(13,22)(14,23)(15,18)(16,33)(19,31)(21,29)(24,40)(25,26)(27,39)(28,38)(30,35)(32,36)(34,37)Polycyclic presentation:
2^8|a^2=ef,b^2=e,c^2=h,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,g^b=gh,f^c=fh,g^c=gh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125732, #125946, #126350, … |
| maximal normal subgroup of | #125732, #125946, #126350, … |
| maximal quotient of | #345674, #345677, #345681, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24247);