Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #30490 |
| automorphism | Aut(G) order 16384 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C24×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C22 | D42 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C22 ◁ C22×C4 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
534 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 534 conjugacy classes of subgroups — too many to draw. Only its 115 normal subgroups are shown.
Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.
| minimal degree | μ(G) = 20 |
| transitive degree | μt(G) = 32 |
A minimal faithful permutation representation — 4 generators on 20 points:
(1,2)(3,4)(5,6,12,16,11,7,10,15)(8,13,19,20,14,9,18,17);(5,7,11,6)(8,17,14,20)(9,18,13,19)(10,15,12,16);(2,4)(5,8)(6,13)(7,9)(10,18)(11,14)(12,19)(15,17)(16,20);(2,4)(5,9)(6,14)(7,8)(10,17)(11,13)(12,20)(15,18)(16,19)Polycyclic presentation:
2^8|a^2=eh,b^2=h,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,g^b=gh,g^c=gh,e^d=eh34 conjugacy classes — view the full character table.
| maximal subgroup of | #153159, #153160, #153167, … |
| maximal normal subgroup of | #153159, #153160, #153167, … |
| maximal quotient of | #332985, #332991, #332998, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 23480);