Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24292 |
| 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 | C22×C4 | C24 |
| prime radical | C2×#41 | C23 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
339 conjugacy classes of subgroups.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2,4,6)(3,7,8,5)(9,10)(11,21)(12,17)(13,23)(14,19)(15,20)(16,22)(18,24);(9,11,14,22,15,12,24,23)(10,16,18,21,20,13,19,17);(1,3,4,8)(2,5,6,7)(9,12)(10,17)(11,15)(13,19)(14,23)(16,18)(20,21)(22,24);(9,13,14,17,15,16,24,21)(10,12,18,23,20,11,19,22)Polycyclic presentation:
2^8|a^2=f,b^2=e,c^2=f,d^2=e,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh40 conjugacy classes — view the full character table.
| maximal subgroup of | C2×#26084, #6443500, #6443501, … |
| maximal normal subgroup of | C2×#26084, #6443500, #6443501, … |
| maximal quotient of | #305865, #305866, #305869, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19072);