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 |
| prime radical | #3354 | C2 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
312 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,6,8)(3,7,5,4)(9,10)(11,18)(12,16)(13,21)(14,17)(15,20)(19,24)(22,23);(2,4)(3,5)(7,8)(11,19)(15,23)(18,24)(20,22);(1,3,6,5)(2,7,8,4)(9,11)(10,15)(12,20)(13,22)(14,19)(16,24)(17,23)(18,21);(1,4)(2,3)(5,8)(6,7)(9,12,14,13)(10,16,17,21)(11,20,19,22)(15,24,23,18)Polycyclic presentation:
2^8|a^2=g,c^2=g,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×#20227, #6359592, #6359648, … |
| maximal normal subgroup of | C2×#20227, #6359592, #6359648, … |
| maximal quotient of | #220903, #220994, #222406, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 13215);