Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #23387 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator, Frattini, cosocle | C22×C4 | C24 |
| LCS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
575 conjugacy classes of subgroups; only the 135 normal subgroups are shown.
This group has 575 conjugacy classes of subgroups — too many to draw. Only its 135 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:
(5,6,8,9)(7,10,11,12)(13,14,16,20,17,21,26,19)(15,22,24,27,23,28,25,18);(1,2)(3,4)(6,9)(10,12)(13,15)(14,18)(16,25)(17,23)(19,22)(20,28)(21,27)(24,26);(2,4)(14,19)(16,26)(18,22)(20,21)(24,25)(27,28);(5,7,8,11)(6,10,9,12)(15,23)(18,27)(22,28)(24,25)Polycyclic presentation:
2^8|a^2=e,d^2=efh,e^2=h,f^2=h,b^a=be,c^a=cf,c^b=cg,d^b=dh,e^b=eh,f^b=fh,e^c=eh,f^c=fh52 conjugacy classes — view the full character table.
| maximal subgroup of | #143154, #143158, #219496, … |
| maximal normal subgroup of | #143154, #143158, #219496, … |
| maximal quotient of | #244135, #244183, #244212, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 16798);