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 | |
571 conjugacy classes of subgroups; only the 115 normal subgroups are shown.
This group has 571 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:
(6,10,13,12)(7,14,16,15)(11,19)(18,20);(1,2)(3,4)(5,6)(7,11)(8,12)(9,13)(10,17)(14,18)(15,20)(16,19);(2,4)(5,7)(6,11)(8,15)(9,16)(10,18)(12,20)(13,19)(14,17);(6,12)(8,17)(10,13)(11,20)(14,15)(18,19)Polycyclic presentation:
2^8|a^2=efh,e^2=h,f^2=h,b^a=be,c^a=cf,d^a=defh,c^b=cg,d^b=de,e^b=eh,f^c=fh,e^d=eh,f^d=fh34 conjugacy classes — view the full character table.
| maximal subgroup of | #125642, #125888, #126295, … |
| maximal normal subgroup of | #125642, #125888, #126295, … |
| maximal quotient of | #343027, #343030, #343034, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 24062);