Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #24264 |
| automorphism | Aut(G) order 32768 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C4×C22⋊C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
371 conjugacy classes of subgroups; only the 139 normal subgroups are shown.
This group has 371 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) = 32 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 32 points:
(1,2)(3,7)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(17,18,21,24)(19,22,27,31)(20,23,28,32)(25,29,26,30);(1,3,4,8)(2,5,6,7)(17,19,20,26,21,27,28,25)(18,22,23,30,24,31,32,29);(9,11,12,16)(10,13,14,15)(19,25)(20,28)(22,29)(23,32)(26,27)(30,31);(2,6)(5,7)Polycyclic presentation:
2^8|a^2=fh,b^2=eg,c^2=f,g^2=h,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh58 conjugacy classes — view the full character table.
| maximal subgroup of | #144593, #144594, C2×#28770, … |
| maximal normal subgroup of | #144593, #144594, C2×#28770, … |
| maximal quotient of | #362134, #364334, #367439, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 21758);