Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #32944 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#121 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
384 conjugacy classes of subgroups; only the 123 normal subgroups are shown.
This group has 384 conjugacy classes of subgroups — too many to draw. Only its 123 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 24 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,5)(4,6)(7,8)(10,14)(11,18)(12,15)(16,20)(17,24)(21,23);(1,3,4,7)(2,5,6,8)(9,10,15,20,13,16,12,14)(11,17,22,24,18,23,19,21);(2,6)(5,8)(10,16)(14,20)(17,23)(21,24);(9,11,12,19,13,18,15,22)(10,17,14,21,16,23,20,24)Polycyclic presentation:
2^8|b^2=efh,d^2=e,e^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,c^b=ch,g^d=gh46 conjugacy classes — view the full character table.
| maximal subgroup of | #6470811, #6470812, #6470813, … |
| maximal normal subgroup of | #6470811, #6470812, #6470813, … |
| maximal quotient of | #351394, #351395, #351400, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 26115);