Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 131072 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C23×C4 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C23 | C22×D4 |
| commutator | C2×C4 | C23×C4 |
| Frattini, cosocle | C22×C4 | C24 |
| prime radical | C24 | C2×D4 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ C22×C42 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
409 conjugacy classes of subgroups; only the 199 normal subgroups are shown.
This group has 409 conjugacy classes of subgroups — too many to draw. Only its 199 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:
(1,2,3,4)(5,6)(7,11)(8,10)(9,12)(13,14)(15,21)(16,18)(17,25)(19,23)(20,24)(22,27)(26,28);(5,7,8,12)(6,9,10,11)(14,18)(15,22)(20,25)(23,28);(7,12)(9,11)(13,15,17,23,16,22,24,28)(14,19,20,21,18,26,25,27);(6,10)(9,11)(14,18)(19,26)(20,25)(21,27)Polycyclic presentation:
2^8|a^2=g,b^2=eh,c^2=f,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce64 conjugacy classes — view the full character table.
| maximal subgroup of | #116607, #116608, #116613, … |
| maximal normal subgroup of | #116607, #116608, #116613, … |
| maximal quotient of | #217365, #217372, #217385, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14948);