Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #18618 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#120 |
| commutator, Frattini, cosocle | C42 | C24 |
| LCS | G ▷ C42 ▷ C22 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C2×C42 ◁ G | |
| LECS | G ▷ C42 ▷ C22 ▷ C1 | |
500 conjugacy classes of subgroups; only the 121 normal subgroups are shown.
This group has 500 conjugacy classes of subgroups — too many to draw. Only its 121 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:
(2,6)(3,4)(7,8)(9,10)(11,19)(12,18)(13,16)(14,17)(15,21)(20,23)(22,24);(1,2,5,8)(3,6,4,7)(9,11,14,20)(10,15,17,24)(12,22,13,21)(16,23,18,19);(1,3,5,4)(2,6,8,7)(10,16)(12,13)(15,23)(17,18)(19,24)(21,22);(1,3,5,4)(2,7,8,6)(10,17)(11,20)(16,18)(21,22)Polycyclic presentation:
2^8|b^2=g,c^2=gh,d^2=gh,e^2=g,f^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,d^b=dg,e^b=eg,f^b=fh,e^c=eh,f^c=fh46 conjugacy classes — view the full character table.
| maximal subgroup of | #250086, #250102, #250110, … |
| maximal normal subgroup of | #250086, #250102, #250110, … |
| maximal quotient of | #166630, #166632, #166638, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 11668);