Finite Group DatabasebetaFinite Group Databasebeta| order | 256 |
| nilpotency class | 3 |
| isoclinism | isoclinic class of #3008 |
| automorphism | Aut(G) order 32768 |
| 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 | C22×D4 | C23 |
| LCS | G ▷ C2×C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C23 ◁ #376 ◁ G | |
| LECS | G ▷ C22×C4 ▷ C2 ▷ C1 | |
431 conjugacy classes of subgroups; only the 171 normal subgroups are shown.
This group has 431 conjugacy classes of subgroups — too many to draw. Only its 171 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:
(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,14)(15,20)(16,26)(17,23)(18,22)(19,24)(21,27)(25,28);(1,3,4,2)(6,9)(10,12)(13,15,19,16)(14,20,24,26)(17,27,18,25)(21,22,28,23);(13,16,19,15)(14,21,24,28)(17,27,18,25)(20,22,26,23);(5,7,8,11)(6,10,9,12)(13,17,19,18)(14,22,24,23)(15,25,16,27)(20,21,26,28)Polycyclic presentation:
2^8|a^2=g,b^2=gh,c^2=h,d^2=eh,f^2=h,b^a=be,c^a=cf,d^a=dh,f^a=fh,d^b=dh,f^c=fh64 conjugacy classes — view the full character table.
| maximal subgroup of | #222849, #222853, #222866, … |
| maximal normal subgroup of | #222849, #222853, #222866, … |
| maximal quotient of | #203199, #203223, #203247, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(256, 14124);