Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #29730 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C23×C4 |
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 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 143 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 143 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 28 |
| transitive degree | μt(G) = 128 |
A minimal faithful permutation representation — 4 generators on 28 points:
(5,6,9,12,8,10,7,11)(13,14)(15,19)(16,20)(17,18)(21,27)(22,25)(23,28)(24,26);(1,2)(3,4)(6,10)(11,12)(14,18)(19,25)(20,26)(27,28);(2,4)(6,11)(7,9)(10,12)(13,15,17,22)(14,19,18,25)(16,23,24,21)(20,28,26,27);(5,7,8,9)(6,11,10,12)(13,16,17,24)(14,20,18,26)(15,21,22,23)(19,27,25,28)Polycyclic presentation:
2^8|a^2=f,c^2=eh,d^2=e,f^2=h,b^a=be,c^a=cf,c^b=cg,d^c=de,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #122546, #122570, #122586, … |
| maximal normal subgroup of | #122546, #122570, #122586, … |
| maximal quotient of | #330737, #330741, #330746, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 22866);