Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23848 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C25 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C2×#127 | C22 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C4⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
507 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 507 conjugacy classes of subgroups — too many to draw. Only its 139 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) = 128 |
A minimal faithful permutation representation — 4 generators on 24 points:
(2,3)(5,6,8,7)(9,10,13,16)(11,14,18,22)(12,15,19,23)(17,21,20,24);(1,2,4,3)(5,7,8,6)(11,17)(12,19)(14,21)(15,23)(18,20)(22,24);(6,7)(9,11,13,18)(10,14,16,22)(12,20,19,17)(15,24,23,21);(1,3,4,2)Polycyclic presentation:
2^8|a^2=fh,b^2=ef,c^2=h,d^2=e,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,g^c=gh52 conjugacy classes — view the full character table.
| maximal subgroup of | #282691, #282708, #282725, … |
| maximal normal subgroup of | #282691, #282708, #282725, … |
| maximal quotient of | #287769, #287770, #287771, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 18357);