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 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
406 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)(3,7)(4,6)(5,8)(9,10)(11,18)(12,23)(13,16)(14,17)(15,20)(19,21)(22,24);(1,3,4,8)(2,5,6,7)(9,10,13,23,14,17,12,16)(11,15,21,24,22,19,20,18);(2,6)(5,7)(9,10)(11,19)(12,16)(13,23)(14,17)(15,22)(18,21)(20,24);(2,6)(5,7)(9,11,12,20,14,22,13,21)(10,15,16,18,17,19,23,24)Polycyclic presentation:
2^8|b^2=eh,d^2=ef,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×#26996, #6443728, #6443730, … |
| maximal normal subgroup of | C2×#26996, #6443728, #6443730, … |
| maximal quotient of | #312689, #312692, #312694, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 19984);