Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29742 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C2×C42 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle, prime radical | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C43 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
291 conjugacy classes of subgroups.
| minimal degree | μ(G) = 40 |
| transitive degree | μt(G) = 256 |
A minimal faithful permutation representation — 4 generators on 40 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,14)(11,15,18,21)(12,16,20,22)(17,23,19,24)(25,26,28,34,30,33,29,32)(27,31,35,38,36,40,37,39);(1,3,4,7)(2,5,6,8)(10,14)(15,21)(16,22)(23,24)(25,27,28,35,30,36,29,37)(26,31,34,38,33,40,32,39);(2,6)(5,8)(9,11,13,18)(10,15,14,21)(12,19,20,17)(16,24,22,23)(26,32)(27,35)(28,29)(33,34)(36,37)(38,39);(9,12,13,20)(10,16,14,22)(11,17,18,19)(15,23,21,24)(26,33)(27,36)(32,34)(35,37)Polycyclic presentation:
2^8|a^2=egh,b^2=f,c^2=e,d^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dh,d^c=de,f^c=fh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | #295171, #295173, #295176, … |
| maximal normal subgroup of | #295171, #295173, #295176, … |
| maximal quotient of | #380218, #380219, #380733, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23222);