Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3186 |
| automorphism | Aut(G) order 262144 |
| homology | Abelianization (H1) = C23×C4 · Schur multiplier (H2) = C25×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 | |
651 conjugacy classes of subgroups; only the 247 normal subgroups are shown.
This group has 651 conjugacy classes of subgroups — too many to draw. Only its 247 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)(7,9)(8,11)(10,12)(13,14)(15,21)(16,19)(17,20)(18,23)(22,27)(24,26)(25,28);(13,15,17,23)(14,18,20,21)(16,24,25,22)(19,27,28,26);(6,9)(7,12)(10,11)(13,16,17,25)(14,19,20,28)(15,22,23,24)(18,26,21,27);(14,20)(18,21)(19,28)(26,27)Polycyclic presentation:
2^8|a^2=g,b^2=e,c^2=e,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=ce,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #107939, #108065, #110217, … |
| maximal normal subgroup of | #107939, #108065, #110217, … |
| maximal quotient of | #215680, #215681, #215682, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14821);