Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #25440 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | #535 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
355 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(9,10,13,16,15,19,14,18)(11,20,21,17,23,12,22,24);(1,3,4,7)(2,5,6,8)(10,16,19,18)(11,21,23,22);(2,6)(5,8)(9,11)(10,17)(12,18)(13,22)(14,21)(15,23)(16,20)(19,24);(9,12,13,24,15,20,14,17)(10,11,18,22,19,23,16,21)Polycyclic presentation:
2^8|a^2=g,b^2=ef,d^2=e,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,f^c=fh,g^c=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#27807, #6457703, #6457708, … |
| maximal normal subgroup of | C2×#27807, #6457703, #6457708, … |
| maximal quotient of | #315013, #315014, #315019, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 20795);