Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24398 |
| automorphism | Aut(G) order 8192 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | C2×#121 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C23×D4 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C23×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
488 conjugacy classes of subgroups; only the 117 normal subgroups are shown.
This group has 488 conjugacy classes of subgroups — too many to draw. Only its 117 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) = 32 |
A minimal faithful permutation representation — 4 generators on 24 points:
(1,2)(3,7)(4,6)(5,8)(9,10,13,16,14,18,12,17)(11,15,19,24,21,22,20,23);(1,3,4,8)(2,5,6,7)(9,11)(10,15)(12,20)(13,19)(14,21)(16,24)(17,23)(18,22);(10,16)(11,19)(12,13)(15,22)(17,18)(20,21);(2,6)(5,7)Polycyclic presentation:
2^8|a^2=fh,b^2=e,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,f^c=fh,g^c=gh43 conjugacy classes — view the full character table.
| maximal subgroup of | #267989, #267991, #268005, … |
| maximal normal subgroup of | #267989, #267991, #268005, … |
| maximal quotient of | #274076, #274081, #274088, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 17509);