Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #20158 |
| 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 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
326 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)(3,8)(6,7)(9,10)(11,18)(12,16)(13,21)(14,17)(15,20)(19,24)(22,23);(1,2)(3,7)(4,6)(5,8)(11,19)(15,23)(18,24)(20,22);(2,6)(5,7)(9,11,14,19)(10,15,17,23)(12,20,13,22)(16,24,21,18);(2,7)(3,8)(5,6)(9,12,14,13)(10,16,17,21)(11,20,19,22)(15,24,23,18)Polycyclic presentation:
2^8|c^2=h,d^2=h,e^2=g,f^2=h,b^a=be,c^a=cf,e^a=eg,f^a=fh,c^b=cgh,d^b=de,e^b=eg,f^c=fh,e^d=eg34 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#20270, #6359643, #6359644, … |
| maximal normal subgroup of | C2×#20270, #6359643, #6359644, … |
| maximal quotient of | #220885, #221011, #222410, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13258);