Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20022 |
| automorphism | Aut(G) order 4096 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C42 | C24 |
| prime radical | #3358 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
327 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,5)(3,7)(6,8)(10,14)(12,20)(15,23)(16,21)(17,24)(18,22);(2,6)(5,8)(9,10,13,16)(11,17,19,23)(12,21,20,14)(15,22,24,18);(1,2)(3,8)(4,6)(5,7);(9,11,13,19)(10,15,16,24)(12,22,20,18)(14,23,21,17)Polycyclic presentation:
2^8|b^2=g,d^2=g,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=ch,d^b=de,e^b=eg,f^c=fh,e^d=eg37 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20032, #6359362, #6359432, … |
| maximal normal subgroup of | C2×#20032, #6359362, #6359432, … |
| maximal quotient of | #222344, #222353, #222572, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13020);